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Dissipation bounds how far biochemical waves wander

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2026-07-09 21:56 UTC pith:T7QKRTUJ

load-bearing objection Clean TUR for traveling wave position diffusion; experimental validation is suggestive but circular the 3 major comments →

arxiv 2607.06997 v1 pith:T7QKRTUJ submitted 2026-07-08 physics.bio-ph nlin.PS

Thermodynamic Limits on Reliable Signaling by Biochemical Traveling Waves

Shengyao Luo , Yuping Chen , Yuansheng Cao This is my paper
classification physics.bio-ph nlin.PS PACS 05.70.Ln87.10.-e87.16.ad
keywords wavewavesdynamicspropagationreliablethermodynamictravelingbiochemical
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Biochemical traveling waves carry signals across cells and tissues, but their positions jitter because the underlying reactions are stochastic. This paper derives a quantitative limit on that jitter: the product of the wave-position diffusion constant and the propagation-specific dissipation rate, divided by the square of the wave speed, must be at least 1. In plain terms, a cell cannot make a traveling wave arrive at a more reproducible time or place without spending more free energy on pushing it forward. The bound is obtained by projecting the full stochastic reaction-diffusion dynamics onto a single collective coordinate, the wave position, which behaves as a biased random walk with drift equal to the wave speed. The key mathematical objects are the Goldstone mode, the shape-invariant translation of the wave profile, and its adjoint partner, the mode that filters stochastic noise into positional wandering. A Cauchy-Schwarz inequality in the noise-weighted inner product between these two modes produces the bound directly. The gap above the bound is controlled by how non-self-adjoint the linearized dynamics are, with two generic sources of deviation: finite wave speed, which breaks spatial symmetry even in single-component systems, and antisymmetric reaction couplings in multi-component systems. For excitable trigger waves, the slow inhibitor field that creates the refractory tail behind the wave dominates the propagation cost, leading to a concrete trade-off among wave speed, inhibitor amplitude, and energy dissipation. The paper tests these predictions in stochastic simulations of a microscopic Belousov-Zhabotinsky reaction-diffusion system and finds consistent signatures in mitotic trigger-wave data from Xenopus egg extracts, where an aging-related decline in energy supply correlates with increased wave-position diffusion. The same bound also limits the maximum reliable firing rate of wave trains, because positional wandering can cause neighboring waves to encroach on each other's refractory zones and annihilate.

Core claim

The central result is the inequality DX times Wprop divided by c-squared is greater than or equal to 1, where DX is the diffusion constant of the wave position, Wprop is the dissipation rate specifically associated with translating the wave profile, and c is the wave speed. This is derived by projecting stochastic reaction-diffusion dynamics onto the adjoint translational mode, which maps the wave position to an effective biased random walk, and then applying Cauchy-Schwarz to the adjoint mode and the Goldstone translational mode in a noise-weighted inner product. The bound is saturated only when the noise-weighted adjoint mode aligns with the translational mode, and the gap is set by thenon

What carries the argument

The derivation projects the stochastic PDE onto the adjoint zero mode psi of the linearized comoving operator L, where L = D*partial_z^2 + c*partial_z + F'(a_bar) and L^dagger psi = 0 with normalization <psi, partial_z a_bar> = 1. This projection yields dX/dt = -<psi, xi>, so the wave position diffuses with DX = <psi, Delta psi>. The propagation dissipation is Wprop = c^2 <partial_z a_bar, Delta^{-1} partial_z a_bar>. Cauchy-Schwarz on the noise-weighted inner product <u,v>_{Delta^{-1}} with the normalization <Delta psi, partial_z a_bar>_{Delta^{-1}} = 1 gives the bound. The gap equals 4 ||(L*)^{-1}_perp L_A partial_z a_bar||^2 / ||partial_z a_bar||^2 in the noise-weighted metric, where L_A,

Load-bearing premise

The entire framework depends on weak-noise linearization: the stochastic dynamics must separate cleanly into a diffusive wave-position coordinate and bounded shape fluctuations. This requires all non-translational eigenmodes of the linearized operator to be stable and the noise to be small enough that the Ornstein-Uhlenbeck approximation holds. Near excitability thresholds, under strong molecular noise, or close to wave breakup, shape modes can destabilize and the reductionto

What would settle it

A direct falsifier would be a stochastic reaction-diffusion system with stable traveling waves where, across a parameter sweep varying dissipation at fixed wave speed, the measured DX * Wprop / c^2 drops below 1. More specifically, if one could construct a system where the noise-weighted adjoint mode and the Goldstone mode are orthogonal rather than aligned, the Cauchy-Schwarz bound would be trivially satisfied but the product could approach zero, which would contradict the claimed lower bound of 1. Another falsifier would be finding that shape fluctuations are not bounded in a regime where

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the bound holds generally, cells transmitting signals via calcium waves, action potentials, or mitotic waves face a fundamental energy-precision trade-off: doubling the timing precision of a wave's arrival requires at least doubling the free energy spent on propagation, all else equal.
  • The annihilation-limited firing rate f_max means that wave-train signaling systems, such as cardiac pacemakers or somitogenesis clocks, have a maximum reliable frequency set jointly by the refractory length and the energy available to suppress positional diffusion, not just by the refractory period alone.
  • The identification of the slow inhibitor as the dominant cost component in excitable waves suggests that evolutionarily tuning inhibitor kinetics, not just activator kinetics, is the primary lever for controlling the energetic efficiency of trigger-wave propagation.
  • The bound provides a benchmark for synthetic biology: engineered reaction-diffusion signaling circuits can be evaluated against a thermodynamic lower bound to assess whether their precision is energy-limited or design-limited.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript derives a thermodynamic uncertainty-type relation for stochastic biochemical traveling waves. The central result, Eq. (4), states that the product of the wave-position diffusion constant D_X and the propagation dissipation rate W_dot_prop, normalized by the square of the wave speed c^2, is bounded below by unity. The bound follows from a Cauchy-Schwarz inequality applied to the adjoint translational mode and the Goldstone mode of the linearized reaction-diffusion operator. The authors then specialize to excitable FitzHugh-Nagumo waves, showing that the slow inhibitor dominates propagation dissipation (Eq. 6), test the predictions in stochastic Belousov-Zhabotinsky simulations with explicit microscopic entropy production, and examine mitotic trigger waves in Xenopus egg extracts. Finally, they derive an annihilation-limited bound on wave-train signaling rates (Eq. 7).

Significance. The paper addresses a genuinely open problem: extending stochastic thermodynamic constraints (analogous to thermodynamic uncertainty relations) to spatially extended, collectively propagating structures. The theoretical construction—projecting noisy field dynamics onto the adjoint translational mode to obtain an effective biased random walk for the wave position—is clean and well-motivated. The Allen-Cahn exact solution (Eq. 49 in SI) provides an analytical check of both the bound and its gap structure, which is a notable strength. The BZ simulations offer a level of independent validation because microscopic entropy production is computed from explicit reaction fluxes rather than from the theory's own predictions. The application to wave-train signaling rates (Eq. 7) is a useful and falsifiable downstream consequence.

major comments (3)
  1. The Xenopus experimental validation (Fig. 4C-D) has a circularity concern that the authors do not fully address. The propagation-dissipation proxy is constructed as the product of wave speed and recovery amplitude (c * A), justified by Eq. (6): W_dot_prop ~ c * omega_m * a_h * m_h / Delta_m. But Eq. (6) is itself a prediction of the same theoretical framework being tested. The 'independent' wavefront-activity proxy (SiR-tubulin maximal rising slope) is correlated with this theory-derived proxy (Fig. 4C), but this correlation only demonstrates internal consistency. The claim in Fig. 4D that 'higher energetic activity is associated with reduced wave-position diffusion, consistent with the thermodynamic bound' therefore does not constitute an independent test of Eq. (4); it tests whether the theory's own scaling relation (Eq. 6) is self-consistent with the bound. The authors should reframe:
  2. The weak-noise linearization that separates wave dynamics into a diffusive translational mode X(t) and bounded shape fluctuations u(z,t) (Eq. 7 in SI) is the weakest load-bearing assumption. The decomposition requires all non-Goldstone eigenvalues of L to have negative real parts and noise sufficiently weak for the Ornstein-Uhlenbeck approximation. The Xenopus extract operates in a regime where noise strength and proximity to excitability boundaries are not independently controlled or measured. While the Discussion acknowledges this limitation, the experimental section does not discuss whether the observed wave statistics (e.g., amplitude variability shown in Fig. S8) are consistent with the weak-noise regime. The authors should comment on whether the observed amplitude fluctuations are small enough for the linearization to be trustworthy, or whether the qualitative agreement in Fig. 4D-
  3. The separation of total dissipation into W_dot_prop and W_dot_int (Eq. 20 in SI) relies on the vanishing of the cross term W_dot_cross at leading order (Eq. 25 in SI), which holds because <u>_ss = 0 within the centered linear mode approximation. The authors note that 'corrections would come from nonlinear shape-mode statistics.' However, for the BZ simulations, the magnitude of this cross term relative to W_dot_prop is not reported. Since the BZ system has more than two species and nonlinear kinetics, it would strengthen the paper to verify that W_dot_cross is indeed negligible in the simulated parameter regime, or at least to bound its magnitude.
minor comments (5)
  1. The notation for the diffusion matrix D (real-space diffusion) and D_X (wave-position diffusion constant) could be confused. Consider distinguishing more clearly, e.g., using D_chem for the former.
  2. In the main text, the definition of the antisymmetric operator L_A (below Eq. 4) uses notation that is dense. Expanding the definition of the hat-F'_A term explicitly for the two-component FHN case would help readers verify the non-reciprocal coupling claim.
  3. Fig. 3C: the axes are labeled 'Normalized diffusion constant' and 'Propagation dissipation rate' but the units and normalization factors are not specified in the caption. Clarify what 'baseline-subtracted' means operationally.
  4. The temporal phase boundaries for the Xenopus data (frames 120, 180, 240, ...) are stated as ad hoc choices. A brief justification for why these particular boundaries yield quasi-stationary windows would be helpful.
  5. Reference [20] is cited for the chemostatted reservoir construction in the BZ model, but the specific choice of uniform reverse rate k^- = 0.5 for all reactions deserves brief physical justification.

Circularity Check

0 steps flagged

Xenopus experimental validation uses a theory-derived dissipation proxy, creating partial circularity in the experimental test of Eq. (4)

full rationale

The central theoretical bound (Eq. 4) is mathematically sound and non-circular: it follows from Cauchy-Schwarz applied to independently defined quantities (DX from noise projection onto adjoint mode, Wprop from convective flux decomposition). The Allen-Cahn exact solution and BZ stochastic simulations provide genuine independent validation, since microscopic entropy production is computed from explicit reaction fluxes (Eqs. 64-67) and DX is measured from ensemble statistics without invoking the theory's predictions. However, the Xenopus experimental validation (Fig. 4D) exhibits partial circularity: because microscopic dissipation cannot be measured, the paper constructs a propagation-cost proxy from the product of wave speed and recovery amplitude (c * A), justified by Eq. (6) — which is itself a prediction of the same theoretical framework being tested. The paper acknowledges this is a proxy, not a direct measurement, and attempts to validate it via correlation with an independent wavefront-activity proxy (SiR-tubulin rising slope, Fig. 4C). But this correlation only shows internal consistency between two empirical quantities, not independent confirmation of the bound. The claim that Fig. 4D is 'consistent with the thermodynamic bound in Eq. (4)' is therefore a test of whether the theory's own scaling relation (Eq. 6) holds, not an independent test of the bound itself. This is a FITTED INPUT CALLED PREDICTION pattern at the experimental validation level, not at the theoretical derivation level. The theory itself remains self-contained and independently validated by the BZ simulations. The circularity is confined to the biological experiment and is openly disclosed by the authors as a limitation ('the thermodynamic quantities are not measured directly'), making this a moderate rather than severe circularity. The score of 4 reflects that the central theoretical claim has independent content and validation, but one of the two experimental validations reduces partially to a theory-derived input.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 2 invented entities

The framework introduces no new physical entities or forces. Wprop and fmax are derived quantities within standard stochastic thermodynamics. The free parameters are simulation choices, not fitted constants. The axioms are standard domain assumptions for reaction-diffusion systems; the weak-noise assumption is the most restrictive.

free parameters (3)
  • beta (Allen-Cahn gap coefficient) = 2(π²-6)/3 ≈ 2.58
    Derived analytically, not fitted to data. Model-dependent constant in the near-equilibrium expansion.
  • BZ reverse rate k- = 0.5
    Set by hand in simulations to render reactions reversible. Not a fitted parameter in the sense of data fitting, but an arbitrary choice affecting absolute dissipation values.
  • Xenopus temporal phase boundaries = 120,180,240,300,360,420,480 frames
    Chosen to define quasi-stationary windows. Not optimized to a result but are post-hoc divisions of the time series.
axioms (4)
  • domain assumption Weak noise regime: fluctuations are small enough that linearization around the traveling wave profile is valid and all non-Goldstone modes are stable.
    Invoked throughout the derivation, starting at the decomposition a = a_bar(z-X(t)) + u(z,t). This is the foundational assumption for the entire framework.
  • domain assumption Spatially homogeneous white noise with diagonal covariance matrix Delta.
    Assumed in the main text Eq. 1 and used for the Cauchy-Schwarz step. Generalized to correlated noise in SI but all numerical/experimental tests use the white noise form.
  • domain assumption One spatial dimension.
    Stated explicitly as a limitation. The entire derivation is 1D. Extension to higher dimensions with curvature and roughening is left as future work.
  • domain assumption Stable traveling wave solution exists with well-defined speed c and profile a_bar(z).
    Required for the Goldstone mode construction and the comoving frame analysis.
invented entities (2)
  • Propagation dissipation rate Wprop independent evidence
    purpose: Separates the dissipation cost of translating the wave profile from internal/housekeeping dissipation.
    Defined operationally via convective flux decomposition (Eq. 15, SI) and validated by baseline subtraction in BZ simulations. Not a new physical entity but a new decomposition of a known quantity.
  • Annihilation-limited firing rate fmax independent evidence
    purpose: Upper bound on reliable wave-train signaling frequency.
    Derived from combining the diffusion bound with a geometric refractory-length constraint. Makes falsifiable predictions for neuronal vs. intracellular waves (Eq. 7).

pith-pipeline@v1.1.0-glm · 26002 in / 2841 out tokens · 278880 ms · 2026-07-09T21:56:49.320799+00:00 · methodology

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read the original abstract

Biochemical traveling waves transmit signals across cells and tissues, but the thermodynamic cost of reliable propagation remains unclear. We develop a stochastic thermodynamic framework for reaction--diffusion systems with stable traveling waves and show that diffusion of the wave position is bounded by the dissipation specifically associated with propagation. The bound follows by projecting noisy field dynamics onto the adjoint translational mode, which maps the wave position to an effective biased random walk. Its tightness is controlled by the non-self-adjoint part of the linearized dynamics, with finite wave speed and antisymmetric reaction dynamics generically producing deviations from equality. For excitable trigger waves in a FitzHugh--Nagumo model, we show that the slow inhibitor dominates the propagation cost, yielding a trade-off among wave speed, inhibitor amplitude, and dissipation. We test these predictions in stochastic simulations of a microscopic Belousov--Zhabotinsky reaction--diffusion system and find consistent signatures in mitotic trigger-wave experiments in \textit{Xenopus} egg extracts. The same relation further imposes an annihilation-limited bound on the reliable signaling rate of wave trains.

Figures

Figures reproduced from arXiv: 2607.06997 by Shengyao Luo, Yuansheng Cao, Yuping Chen.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

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Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    J. B. Chang and J. E. Ferrell Jr, Nature 500, 603 (2013)

    work page 2013

  2. [2]

    S. E. Brantley and S. Di Talia, Development 148, dev193128 (2021)

    work page 2021

  3. [3]

    Gelens, G

    L. Gelens, G. A. Anderson, and J. E. Ferrell Jr, Molec- ular biology of the cell 25, 3486 (2014)

    work page 2014

  4. [4]

    Huang, Y

    J.-H. Huang, Y. Chen, W. Y. Huang, S. Tabatabaee, and J. E. Ferrell Jr, Nature Communications 15, 5782 (2024). 7

    work page 2024

  5. [5]

    Cooke and E

    J. Cooke and E. C. Zeeman, Journal of theoretical biology 58, 455 (1976)

    work page 1976

  6. [6]

    Aulehla, W

    A. Aulehla, W. Wiegraebe, V. Baubet, M. B. Wahl, C. Deng, M. Taketo, M. Lewandoski, and O. Pourquié, Nature cell biology 10, 186 (2008)

    work page 2008

  7. [7]

    A. C. Oates, L. G. Morelli, and S. Ares, Development 139, 625 (2012)

    work page 2012

  8. [8]

    Vergassola, V

    M. Vergassola, V. E. Deneke, and S. Di Talia, Proceed- ings of the National Academy of Sciences 115, E2165 (2018)

    work page 2018

  9. [9]

    Di Talia and M

    S. Di Talia and M. Vergassola, Annual review of bio- physics 51, 327 (2022)

    work page 2022

  10. [10]

    Leybaert and M

    L. Leybaert and M. J. Sanderson, Physiological reviews 92, 1359 (2012)

    work page 2012

  11. [11]

    N. Kuga, T. Sasaki, Y. Takahara, N. Matsuki, and Y. Ikegaya, Journal of Neuroscience 31, 2607 (2011)

    work page 2011

  12. [12]

    N. Hino, L. Rossetti, A. Marín-Llauradó, K. Aoki, X. Trepat, M. Matsuda, and T. Hirashima, Develop- mental cell 53, 646 (2020)

    work page 2020

  13. [13]

    Bolado-Carrancio, O

    A. Bolado-Carrancio, O. S. Rukhlenko, E. Nikonova, M. A. Tsyganov, A. Wheeler, A. Garcia-Munoz, W. Kolch, A. von Kriegsheim, and B. N. Kholodenko, Elife 9, e58165 (2020)

    work page 2020

  14. [14]

    Whitaker, Physiological reviews 86, 25 (2006)

    M. Whitaker, Physiological reviews 86, 25 (2006)

    work page 2006

  15. [15]

    B.-K. Liao, D. J. Jörg, and A. C. Oates, Nature Com- munications 7, 11861 (2016)

    work page 2016

  16. [16]

    A. C. Barato and U. Seifert, Physical review letters 114, 158101 (2015)

    work page 2015

  17. [17]

    Y. Cao, H. Wang, Q. Ouyang, and Y. Tu, Nature physics 11, 772 (2015)

    work page 2015

  18. [18]

    T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Physical review letters 116, 120601 (2016)

    work page 2016

  19. [19]

    J. M. Horowitz and T. R. Gingrich, Physical Review E 96, 020103 (2017)

    work page 2017

  20. [20]

    A vanzini, G

    F. A vanzini, G. Falasco, and M. Esposito, The Journal of Chemical Physics 151 (2019)

    work page 2019

  21. [21]

    J. M. Horowitz and T. R. Gingrich, Nature Physics 16, 15 (2020)

    work page 2020

  22. [22]

    Marsland, W

    R. Marsland, W. Cui, and J. M. Horowitz, Journal of the Royal Society Interface 16 (2019)

    work page 2019

  23. [23]

    Macieszczak, K

    K. Macieszczak, K. Brandner, and J. P. Garrahan, Phys. Rev. Lett. 121, 130601 (2018)

    work page 2018

  24. [24]

    Esposito, Phys

    M. Esposito, Phys. Rev. E 85, 041125 (2012)

    work page 2012

  25. [25]

    Kawaguchi and Y

    K. Kawaguchi and Y. Nakayama, Phys. Rev. E 88, 022147 (2013)

    work page 2013

  26. [26]

    Rao and M

    R. Rao and M. Esposito, Phys. Rev. X 6, 041064 (2016)

    work page 2016

  27. [27]

    J. M. Davidenko, A. V. Pertsov, R. Salomonsz, W. Bax- ter, and J. Jalife, Nature 355, 349 (1992)

    work page 1992

  28. [28]

    A. M. Pertsov, J. M. Davidenko, R. Salomonsz, W. T. Baxter, and J. Jalife, Circulation research 72, 631 (1993)

    work page 1993

  29. [29]

    Radivojevic, F

    M. Radivojevic, F. Franke, M. Altermatt, J. Müller, A. Hierlemann, and D. J. Bakkum, Elife 6, e30198 (2017)

    work page 2017

  30. [30]

    Yang and M

    Y. Yang and M. Wu, Philosophical Transactions of the Royal Society B: Biological Sciences 373 (2018)

    work page 2018

  31. [31]

    Gerhardt, M

    M. Gerhardt, M. Ecke, M. Walz, A. Stengl, C. Beta, and G. Gerisch, Journal of cell science 127, 4507 (2014)

    work page 2014

  32. [32]

    O. D. Weiner, W. A. Marganski, L. F. Wu, S. J. Altschuler, and M. W. Kirschner, PLoS biology 5, e221 (2007)

    work page 2007

  33. [33]

    Y. Miao, S. Bhattacharya, T. Banerjee, B. Abubaker- Sharif, Y. Long, T. Inoue, P. A. Iglesias, and P. N. Devreotes, Molecular systems biology 15, MSB188585 (2019)

    work page 2019

  34. [34]

    Niggemann and U

    O. Niggemann and U. Seifert, Journal of Statistical Physics 178, 1142 (2020)

    work page 2020

  35. [35]

    Niggemann and U

    O. Niggemann and U. Seifert, Journal of Statistical Physics 186, 3 (2022)

    work page 2022

  36. [36]

    Nardini, É

    C. Nardini, É. Fodor, E. Tjhung, F. Van Wijland, J. Tailleur, and M. E. Cates, Physical Review X 7, 021007 (2017)

    work page 2017

  37. [37]

    Markovich, É

    T. Markovich, É. Fodor, E. Tjhung, and M. E. Cates, Physical Review X 11, 021057 (2021)

    work page 2021

  38. [38]

    Liang, P

    S. Liang, P. De Los Rios, and D. M. Busiello, Physical review letters 132, 228402 (2024)

    work page 2024

  39. [39]

    M. C. Cross and P. C. Hohenberg, Reviews of modern physics 65, 851 (1993)

    work page 1993

  40. [40]

    J. J. Tyson and J. P. Keener, Physica D: Nonlinear Phe- nomena 32, 327 (1988)

    work page 1988

  41. [41]

    Lindner, J

    B. Lindner, J. Garcıa-Ojalvo, A. Neiman, and L. Schimansky-Geier, Physics reports 392, 321 (2004)

    work page 2004

  42. [42]

    J. P. Keener and T. J. Lewis, SIAM Journal on Applied Mathematics 61, 293 (2000)

    work page 2000

  43. [43]

    J. Yang, F. Xie, Z. Qu, and A. Garfinkel, Physical review letters 91, 148302 (2003)

    work page 2003

  44. [44]

    F. H. Fenton, E. M. Cherry, H. M. Hastings, and S. J. Evans, Chaos: An Interdisciplinary Journal of Nonlinear Science 12, 852 (2002)

    work page 2002

  45. [45]

    J. P. Keener, SIAM Journal on Applied Mathematics 46, 1039 (1986)

    work page 1986

  46. [46]

    Panja, Physics Reports 393, 87 (2004)

    D. Panja, Physics Reports 393, 87 (2004)

    work page 2004

  47. [47]

    Meron, Physics reports 218, 1 (1992)

    E. Meron, Physics reports 218, 1 (1992)

    work page 1992

  48. [48]

    Thermodynamic Limits on Reliable Signaling by Biochemical Traveling Waves

    S. J. Bryant and B. B. Machta, Physical review letters 131, 068401 (2023). Supporting Information for “Thermodynamic Limits on Reliable Signaling by Biochemical Traveling Waves” FIELD METHOD FOR TRA VELING W A VE SYSTEM We consider an n-component continuous field a(x, t) ∈ Rn whose spatiotemporal evolution is governed by the following stochastic partial d...

    work page 2023