pith. sign in

arxiv: 2512.15737 · v3 · submitted 2025-12-06 · 🧬 q-bio.MN

Enzyme-Substrate Complex Formation Modulates Diffusion-Driven Patterning In Metabolic Pathways

Faezeh Farivar This is my paper

Pith reviewed 2026-05-17 01:13 UTC · model grok-4.3

classification 🧬 q-bio.MN
keywords Turing instabilityenzyme-substrate complexreaction-diffusionmetabolic pathwayspattern formationquasi-steady-statediffusion-driven instability
0
0 comments X

The pith

Reversible enzyme-substrate binding modifies the conditions for diffusion-driven pattern formation in metabolic pathways.

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

In a two-step metabolic pathway, enzymes bind reversibly to their substrates before catalysis. The paper models this explicitly as a three-species reaction-diffusion system and reduces it to two variables by assuming the complex reaches equilibrium quickly. Linear stability analysis then shows that this binding changes where and how easily spatial patterns emerge from diffusion instabilities, unlike models that use simplified rate laws. A reader might care because many cellular processes involve such binding and could lead to organized metabolite distributions without physical compartments.

Core claim

Starting from a mechanistic model that includes reversible enzyme-substrate complex formation, the analysis demonstrates that the quasi-steady-state reduced system exhibits a shifted Turing instability region compared to effective kinetics models. Weakly nonlinear analysis and simulations confirm that binding interactions alter pattern selection and slow the development of spatial heterogeneity in metabolite concentrations.

What carries the argument

The quasi-steady-state approximation applied to the enzyme-substrate complex, which eliminates the fast binding variable while retaining the nonlinear effects on the reaction terms.

If this is right

  • The location and size of the parameter region allowing diffusion-driven instability depend on the binding and unbinding rates.
  • Reversible binding can either promote or suppress pattern formation relative to irreversible or effective models.
  • Pattern emergence is slower when enzyme-substrate interactions are accounted for explicitly.
  • These changes provide a mechanistic explanation for mesoscale metabolic organization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If binding kinetics vary across enzymes, different metabolic steps could have distinct pattern-forming tendencies.
  • This framework could be extended to longer pathways to see how binding propagates spatial structure.
  • Experimental tests might involve varying enzyme concentrations or using mutants with altered binding affinities to observe changes in metabolite clustering.

Load-bearing premise

The binding and unbinding of substrate to enzyme occur on a much faster time scale than changes in the overall metabolite concentrations.

What would settle it

A direct measurement of the critical diffusion coefficients or reaction rates at which patterns first appear in a controlled metabolic system that does not match the boundaries predicted by the stability analysis of the reduced model.

Figures

Figures reproduced from arXiv: 2512.15737 by Faezeh Farivar.

Figure 1
Figure 1. Figure 1: Phase-plane comparison of the simplified and non￾degenerated enzyme-pathway models. Arrows represent vector fields (dark gray: primary model; light gray: non-degenerated model). Red/blue and magenta/green curves denote the corresponding nullclines, whose intersections yield homogeneous steady states (dots). Trajectories from identical initial conditions are shown in magenta and green. The non￾degenerated f… view at source ↗
Figure 2
Figure 2. Figure 2: Plots of polynomials B(k 2 ) of two models (3.3) (red plots) and (1.1) (black plots) (right figure). Plots of dispersion relation poly￾nomials λ 2 (k 2 ) (left figure). Given parameters are α0 = 0.01(s −1 ), k1 = 0.05(µM−1 s −1 ), k2 = 0.02(µM−1 s −1 ), kd1 = 0.01s −1 , ka1 = 0.03(µM−1 s −1 ), kcat = 0.1s −1 , β = 0.01(s −1 ), C ∗ 0 = 0.1(µMs−1 ), n1 = 10(µM), n2 = 5(µM), γ0 = 0.1(µm2 s −1 ), γ1 = 0.01(µm2… view at source ↗
Figure 3
Figure 3. Figure 3: Variation of Turing regions in plane (n1, d) as the param￾eters D0 (left plot) and D1 (right plot) increase. Red dots in the plots indicate dc threshold for each given D0 or D1. Parameters have chosen like [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Variation of Turing regions in plane (n1, γ) as the parameters γ0 (left plot) and γ1 (right plot) increase. Red dots in the plots indicate γc threshold for each given γ0 or γ1. Parameters have chosen like [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of Turing regions in plane (n1, γ) as the parameter kcat, kd1 , ka1 increases res. Parameters have chosen like [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variation of Turing regions in plane (n1, d) (right plot) and (n1, γ) (left plot) as the parameter β increases. Parameters have chosen like [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Variation of Turing regions in plane (n1, d) (right plot) and (n1, γ) (left) as the parameter α increases. Parameters have chosen like [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Variation of Turing regions in plane (n1, d) (right plot) and (n1, γ) (left plot). Variation of Turing regions in plane (n2, d) and (n2, γ) (right plot). Parameters have chosen like [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Turing Pattern for given parameters: n1 = 3.5, n2 = 1, k1 = k2 = 2, α0 = 1, C∗ 0 = 7, β = 3, D0 = 2, D1 = 2.5 that provides C E 0 = 0.87, CE 1 = 1.22, kc = 2.8, dc = 9.51. u 0 (x,t) 0 2 4 6 x 0 500 1000 1500 t 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 u 1 (x,t) 0 2 4 6 x 0 500 1000 1500 t 0.9 0.95 1 1.05 1.1 1.15 1.2 [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Turing Pattern for given parameters: n1 = 3.5, n2 = 1, ka1 = 5, kcat = 8, kd1 = 2.5; k2 = 2, α0 = 1, C∗ 0 = 7, β = 3, D0 = 2, D1 = 2.5 that provides u E 0 = 0.44, uE 1 = 1.19, kc = 2.5, dc = 14.76. an explicit enzyme–substrate interaction, in which S0 must first bind to the enzyme E1 to form the complex E1S0 before being converted into the intermediate S1. This reversible binding acts as a biochemical buf… view at source ↗
Figure 11
Figure 11. Figure 11: Numerical results vs WNL of model 1 for given parameters: as [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Numerical results vs WNL of model 2 for given parameters: as [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
read the original abstract

Spatial organization in metabolic pathways can arise from the interplay between enzymatic reaction kinetics and diffusion-driven instabilities. In this work we investigate how reversible enzyme--substrate binding influences pattern formation in a two-step metabolic pathway. Starting from a mechanistic description in which the substrate reversibly binds to the first enzyme before catalytic conversion, we formulate a three-species reaction--diffusion system that explicitly incorporates the enzyme--substrate complex. We first analyse the homogeneous dynamics and determine the unique steady state of the kinetic system. Exploiting the separation of time scales between the rapid binding kinetics and the slower evolution of metabolite concentrations, we derive a reduced two-variable model using a quasi-steady-state approximation for the enzyme-substrate complex. This reduction preserves the essential nonlinear coupling between catalytic reactions and spatial transport. Linear stability and weakly nonlinear analysis reveal conditions for diffusion-driven (Turing) instability and show that reversible enzyme binding significantly modifies the location and extent of the instability region compared to models with effective kinetics. Numerical simulations confirm the analytical predictions and demonstrate how enzyme-substrate interactions reshape pattern selection and slow the emergence of spatial heterogeneity. These results provide a mechanistic link between enzyme binding kinetics, diffusion-driven pattern formation, and mesoscale metabolic organization. The proposed framework offers a tractable approach for studying spatial patterning in enzymatic networks and may help explain the emergence of structured biochemical domains such as those associated with liquid--liquid phase separation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a three-species reaction-diffusion model for a two-step metabolic pathway that explicitly includes reversible enzyme-substrate complex formation. After identifying the unique homogeneous steady state, the authors invoke a quasi-steady-state approximation (QSSA) on the complex to obtain a reduced two-variable system. Linear stability analysis together with weakly nonlinear analysis then identifies the conditions for diffusion-driven (Turing) instability and demonstrates that reversible binding shifts the location and extent of the instability region relative to models that employ effective kinetics. Numerical simulations are used to corroborate the analytical predictions and to illustrate effects on pattern selection and the time scale of heterogeneity emergence.

Significance. If the central results are robust, the work supplies a concrete mechanistic link between enzyme-binding kinetics and the parameter regimes that permit spatial patterning in metabolic pathways. The explicit retention of binding steps, rather than immediate reduction to effective rates, is a clear strength; the combination of linear stability, weakly nonlinear analysis, and direct numerical confirmation provides a tractable analytical route that could be applied to larger enzymatic networks. The findings offer a plausible explanation for mesoscale metabolic organization without invoking additional regulatory mechanisms.

major comments (2)
  1. [§3] §3 (QSSA reduction): The reduction assumes that binding equilibrates much faster than diffusion and metabolite evolution, allowing the complex to be slaved locally. However, the manuscript provides no explicit comparison of the dispersion relation or the critical diffusion ratio between the full three-species system and the reduced two-variable model. Without this comparison or an error bound on the approximation, it remains unclear whether the reported shifts in the instability region are preserved when diffusion of the complex is retained.
  2. [§4] §4 (linear stability analysis): The claim that reversible enzyme binding 'significantly modifies' the location and extent of the Turing region is central, yet the text does not quantify the magnitude of the shift (e.g., change in critical wave number or minimal diffusion ratio) for representative parameter values. A direct overlay of the instability boundaries for the reduced model versus the effective-kinetics model would make the modification concrete and testable.
minor comments (2)
  1. Notation for the binding and unbinding rates is introduced without a consolidated table of symbols; adding such a table would improve readability when comparing the full and reduced systems.
  2. The weakly nonlinear analysis section would benefit from an explicit statement of the amplitude equation coefficients and the conditions under which they remain positive, to allow readers to reproduce the pattern-selection conclusions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the positive assessment of the significance of our work and address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (QSSA reduction): The reduction assumes that binding equilibrates much faster than diffusion and metabolite evolution, allowing the complex to be slaved locally. However, the manuscript provides no explicit comparison of the dispersion relation or the critical diffusion ratio between the full three-species system and the reduced two-variable model. Without this comparison or an error bound on the approximation, it remains unclear whether the reported shifts in the instability region are preserved when diffusion of the complex is retained.

    Authors: We agree with the referee that an explicit comparison would better justify the QSSA. In the revised version, we will add a section comparing the dispersion relations of the full three-species system and the reduced two-variable model for parameters where the binding time scale is sufficiently separated from diffusion. We will also derive and present error bounds on the approximation to confirm that the reported shifts in the Turing instability region remain valid. revision: yes

  2. Referee: [§4] §4 (linear stability analysis): The claim that reversible enzyme binding 'significantly modifies' the location and extent of the Turing region is central, yet the text does not quantify the magnitude of the shift (e.g., change in critical wave number or minimal diffusion ratio) for representative parameter values. A direct overlay of the instability boundaries for the reduced model versus the effective-kinetics model would make the modification concrete and testable.

    Authors: We acknowledge that providing quantitative measures and a visual comparison would strengthen the central claim. We will revise the manuscript to include an overlay plot of the instability boundaries in the (diffusion ratio, wave number) plane for both the reduced model with explicit binding and the effective-kinetics model. Additionally, we will report specific numerical values for the shifts in critical parameters, such as the minimal diffusion ratio required for instability, using representative parameter sets. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no reductions to inputs by construction

full rationale

The paper begins with an explicit three-species mechanistic reaction-diffusion system incorporating reversible enzyme-substrate binding. It invokes the standard quasi-steady-state approximation justified by explicit time-scale separation between fast binding and slow metabolite evolution, yielding a reduced two-variable model whose Jacobian and dispersion relation are then analyzed for Turing instability. The reported modification of the instability region relative to effective-kinetics models is obtained by direct comparison of these derived dispersion relations; it is not a fitted parameter renamed as a prediction, nor does any step reduce to a self-definition or self-citation chain. The derivation remains independent of the target result and is externally falsifiable via the full three-species system or numerical simulation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model depends on standard mass-action kinetics and time-scale separation; no new entities are postulated.

free parameters (2)
  • binding and unbinding rate constants
    Reversible binding rates appear as parameters in the three-species system and are required for the QSSA reduction.
  • diffusion coefficients
    Diffusion rates for substrate, complex, and product are model inputs that control the instability condition.
axioms (2)
  • domain assumption Quasi-steady-state approximation holds for the enzyme-substrate complex
    Invoked to reduce the three-species system to two variables based on rapid binding relative to metabolite evolution.
  • standard math Mass-action kinetics govern the reactions
    Standard assumption for formulating the reaction-diffusion equations.

pith-pipeline@v0.9.0 · 5549 in / 1411 out tokens · 66314 ms · 2026-05-17T01:13:21.920155+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 1 internal anchor

  1. [1]

    Bending the rules: curvature’s impact on cell biology.BMC biology, 23(1):296, 2025

    Carole Arnold, Ismail Tahmaz, Marie-Ly Chapon, Hasna Maayouf, Valeriy Luchnikov, Jean-Louis Milan, Fabricio Borghi, and Laurent Pieuchot. Bending the rules: curvature’s impact on cell biology.BMC biology, 23(1):296, 2025. 4 MODEL OF AN ENZYME 25

    work page 2025

  2. [2]

    Biomolecular condensates: organizers of cellular biochemistry.Nature reviews Molecular cell biology, 18(5):285–298, 2017

    Salman F Banani, Hyun O Lee, Anthony A Hyman, and Michael K Rosen. Biomolecular condensates: organizers of cellular biochemistry.Nature reviews Molecular cell biology, 18(5):285–298, 2017. 4

    work page 2017

  3. [3]

    Enzyme activity regulates sub- stratediffusionbymodulatingviscosityincrowdedmilieu.BioRxiv, pages2024–09,

    Alessandro Bevilacqua, Mauricio Rios Maciel, Mark V Sullivan, Stefano Pascarelli, Mirco Dindo, Amy Q Shen, and Paola Laurino. Enzyme activity regulates sub- stratediffusionbymodulatingviscosityincrowdedmilieu.BioRxiv, pages2024–09,

    work page

  4. [4]

    Clustering and optimal arrangement of enzymes in reaction-diffusion systems.Physical Review Letters, 110(20):208104, 2013

    Alexander Buchner, Filipe Tostevin, and Ulrich Gerland. Clustering and optimal arrangement of enzymes in reaction-diffusion systems.Physical Review Letters, 110(20):208104, 2013. 2

    work page 2013

  5. [5]

    Enzyme clustering accel- erates processing of intermediates through metabolic channeling.Nature biotech- nology, 32(10):1011–1018, 2014

    Michele Castellana, Maxwell Z Wilson, Yifan Xu, Preeti Joshi, Ileana M Cristea, Joshua D Rabinowitz, Zemer Gitai, and Ned S Wingreen. Enzyme clustering accel- erates processing of intermediates through metabolic channeling.Nature biotech- nology, 32(10):1011–1018, 2014. 2, 4, 5

    work page 2014

  6. [6]

    Liquid–liquid phase separation: a potentially fundamental mechanism of sepsis.Cell Death Discovery, 11(1):310,

    Huiyi Chen, Shunyi Huang, Longcheng Quan, Caiyuan Yu, Yang Zhu, Xiaocong Sun, Yuanli Zhang, Liehua Deng, and Feng Chen. Liquid–liquid phase separation: a potentially fundamental mechanism of sepsis.Cell Death Discovery, 11(1):310,

    work page

  7. [7]

    Liquid–liquid phase separation of biomacromolecules and its roles in metabolic diseases.Cells, 11(19):3023, 2022

    Zhihao Chen, Ying Huai, Wenjing Mao, Xuehao Wang, Kang Ru, Airong Qian, and Hong Yang. Liquid–liquid phase separation of biomacromolecules and its roles in metabolic diseases.Cells, 11(19):3023, 2022. 2

    work page 2022

  8. [8]

    PhD thesis, Johns Hopkins University, 2022

    Basilio Cieza Huaman.Investigation of Molecular condensates using a Reaction- Diffusion Master Equation Model. PhD thesis, Johns Hopkins University, 2022. 2

    work page 2022

  9. [9]

    Chemical oscillators in structured media

    Irving R Epstein, Vladimir K Vanag, Anna C Balazs, Olga Kuksenok, Pratyush Dayal, and Amitabh Bhattacharya. Chemical oscillators in structured media. Accounts of chemical research, 45(12):2160–2168, 2012. 2

    work page 2012

  10. [10]

    Emergence of multicellular organisms with dynamic differentiation and spatial pattern.Artificial life, 4(1):79–93, 1998

    Chikara Furusawa and Kunihiko Kaneko. Emergence of multicellular organisms with dynamic differentiation and spatial pattern.Artificial life, 4(1):79–93, 1998. 2

    work page 1998

  11. [11]

    Effects of cross-diffusion on Turing patterns in a reaction-diffusion Schnakenberg model

    G Gambino, S Lupo, and M Sammartino. Effects of cross-diffusion on turing pat- ternsinareaction-diffusionschnakenbergmodel.arXiv preprint arXiv:1501.04890,

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross- diffusion.Mathematics and Computers in Simulation, 82(6):1112–1132, 2012

    Gaetana Gambino, Maria Carmela Lombardo, and Marco Sammartino. Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross- diffusion.Mathematics and Computers in Simulation, 82(6):1112–1132, 2012. 18

    work page 2012

  13. [13]

    Pattern 26 F

    Gaetana Gambino, Maria Carmela Lombardo, and Marco Sammartino. Pattern 26 F. F ARIV AR formation driven by cross-diffusion in a 2d domain.Nonlinear Analysis: Real World Applications, 14(3):1755–1779, 2013. 6

    work page 2013

  14. [14]

    Pattern selection in the 2d fitzhugh–nagumo model.Ricerche di Matem- atica, 68(2):535–549, 2019

    Gaetana Gambino, Maria Cristina Lombardo, Gino Rubino, and Marcello Sam- martino. Pattern selection in the 2d fitzhugh–nagumo model.Ricerche di Matem- atica, 68(2):535–549, 2019. 6

    work page 2019

  15. [15]

    Pattern for- mationandtransitiontochaosinachemotaxismodelofacuteinflammation.SIAM Journal on Applied Dynamical Systems, 20(4):1844–1881, 2021

    Valeria Giunta, Maria Carmela Lombardo, and Marco Sammartino. Pattern for- mationandtransitiontochaosinachemotaxismodelofacuteinflammation.SIAM Journal on Applied Dynamical Systems, 20(4):1844–1881, 2021. 2

    work page 2021

  16. [16]

    Nicepath: Finding metabolic pathways in large networks through atom-conserving substrate–product pairs.Bioinformat- ics, 37(20):3560–3568, 2021

    Jasmin Hafner and Vassily Hatzimanikatis. Nicepath: Finding metabolic pathways in large networks through atom-conserving substrate–product pairs.Bioinformat- ics, 37(20):3560–3568, 2021. 2

    work page 2021

  17. [17]

    Metabolomics, standards, and metabolic modeling for synthetic biology in plants.Frontiers in Bioengineering and Biotechnology, 3:167, 2015

    Camilla Beate Hill, Tobias Czauderna, Matthias Klapperstück, Ute Roessner, and Falk Schreiber. Metabolomics, standards, and metabolic modeling for synthetic biology in plants.Frontiers in Bioengineering and Biotechnology, 3:167, 2015. 2

    work page 2015

  18. [18]

    A user’s guide to pde models for chemotaxis

    Thomas Hillen and Kevin J Painter. A user’s guide to pde models for chemotaxis. Journal of mathematical biology, 58(1):183–217, 2009. 4

    work page 2009

  19. [19]

    Liquid-liquid phase separation in biology.Annual review of cell and developmental biology, 30(1): 39–58, 2014

    Anthony A Hyman, Christoph A Weber, and Frank Jülicher. Liquid-liquid phase separation in biology.Annual review of cell and developmental biology, 30(1): 39–58, 2014. 4

    work page 2014

  20. [20]

    Springer, 2009

    James Keener and James Sneyd.Mathematical physiology: II: Systems physiology. Springer, 2009. 5, 6

    work page 2009

  21. [21]

    Prentice hall Upper Saddle River, NJ, 2002

    Hassan K Khalil and Jessy W Grizzle.Nonlinear systems, volume 3. Prentice hall Upper Saddle River, NJ, 2002

    work page 2002

  22. [22]

    A reaction network model of microscale liquid–liquid phase separation reveals effects of spatial dimension.The Journal of Chemical Physics, 161(20), 2024

    Jinyoung Kim, Sean D Lawley, and Jinsu Kim. A reaction network model of microscale liquid–liquid phase separation reveals effects of spatial dimension.The Journal of Chemical Physics, 161(20), 2024. 2

    work page 2024

  23. [23]

    Reentrant liquid condensate phase of proteins is stabilized by hydropho- bic and non-ionic interactions.Biophysical Journal, 120(3):28a, 2021

    Georg Krainer, Timothy J Welsh, Jerelle A Joseph, Peter St George-Hyslop, An- thony A Hyman, Rosana Collepardo-Guevara, Simon Alberti, and Tuomas PJ Knowles. Reentrant liquid condensate phase of proteins is stabilized by hydropho- bic and non-ionic interactions.Biophysical Journal, 120(3):28a, 2021. 2

    work page 2021

  24. [24]

    Biochem- ical evaluation of molecular parts for flavonoid production using plant synthetic biology.Frontiers in Plant Science, 16:1528122, 2025

    Hyo Lee, Saet Buyl Lee, Sangkyu Park, Jaeeun Song, and Beom-Gi Kim. Biochem- ical evaluation of molecular parts for flavonoid production using plant synthetic biology.Frontiers in Plant Science, 16:1528122, 2025. 2

    work page 2025

  25. [25]

    Peptide-mediated liquid–liquid phase separation and biomolecular condensates.Soft Matter, 21(10):1781–1812,

    Guangle Li, Chengqian Yuan, and Xuehai Yan. Peptide-mediated liquid–liquid phase separation and biomolecular condensates.Soft Matter, 21(10):1781–1812,

    work page

  26. [26]

    Phase-separated biomolecular condensates for MODEL OF AN ENZYME 27 biocatalysis.Trends in Biotechnology, 42(4):496–509, 2024

    Samuel Lim and Douglas S Clark. Phase-separated biomolecular condensates for MODEL OF AN ENZYME 27 biocatalysis.Trends in Biotechnology, 42(4):496–509, 2024. 2

    work page 2024

  27. [27]

    Liquid–liquid phase separation: Roles and implications in future cancer treatment.International Journal of Biological Sciences, 19(13):4139, 2023

    Zheran Liu, Zijian Qin, Yingtong Liu, Xi Xia, Ling He, Na Chen, Xiaolin Hu, and Xingchen Peng. Liquid–liquid phase separation: Roles and implications in future cancer treatment.International Journal of Biological Sciences, 19(13):4139, 2023. 2

    work page 2023

  28. [28]

    Emerging roles of liquid–liquid phase separation in cancer: from protein aggregation to immune-associated signaling.Frontiers in cell and devel- opmental biology, 9:631486, 2021

    Jiahua Lu, Junjie Qian, Zhentian Xu, Shengyong Yin, Lin Zhou, Shusen Zheng, and Wu Zhang. Emerging roles of liquid–liquid phase separation in cancer: from protein aggregation to immune-associated signaling.Frontiers in cell and devel- opmental biology, 9:631486, 2021. 2

    work page 2021

  29. [29]

    The turing model for biological pattern formation

    Philip K Maini and Thomas E Woolley. The turing model for biological pattern formation. InThe dynamics of biological systems, pages 189–204. Springer, 2019. 2

    work page 2019

  30. [30]

    Turing’s model for biological pattern formation and the robustness problem.Interface focus, 2(4):487–496, 2012

    Philip K Maini, Thomas E Woolley, Ruth E Baker, Eamonn A Gaffney, and S Seirin Lee. Turing’s model for biological pattern formation and the robustness problem.Interface focus, 2(4):487–496, 2012. 2

    work page 2012

  31. [31]

    Emerging roles for phase separation of rna-binding proteins in cellular pathology of als.Frontiers in cell and developmental biology, 10:840256, 2022

    Katarina Milicevic, Branislava Rankovic, Pavle R Andjus, Danijela Bataveljic, and Dragomir Milovanovic. Emerging roles for phase separation of rna-binding proteins in cellular pathology of als.Frontiers in cell and developmental biology, 10:840256, 2022. 2

    work page 2022

  32. [32]

    A reaction-diffusion model captures the essence of liquid-liquid phase separation

    Nayana Mukherjee, Abdul Wasim, Jagannath Mondal, and Pushpita Ghosh. A reaction-diffusion model captures the essence of liquid-liquid phase separation. bioRxiv, pages 2024–04, 2024. 2

    work page 2024

  33. [33]

    An introduction, volume 17

    James D Murray.Mathematical biology: I. An introduction, volume 17. Springer Science & Business Media, 2007. 6

    work page 2007

  34. [34]

    A time-resolved diffusion technique for detection of the conformational changes and molecular assembly/disassembly processes of biomolecules.Frontiers in Genetics, 12:691010, 2021

    Yusuke Nakasone and Masahide Terazima. A time-resolved diffusion technique for detection of the conformational changes and molecular assembly/disassembly processes of biomolecules.Frontiers in Genetics, 12:691010, 2021. 2

    work page 2021

  35. [35]

    PhD thesis, Politecnico di Torino, 2022

    Gabriele Rigano.Models of cell membrane patterning via diffusion-controlled phase separation. PhD thesis, Politecnico di Torino, 2022. 2

    work page 2022

  36. [36]

    Curvature in biological systems: its quantification, emer- gence, and implications across the scales.advanced materials, 35(13):2206110,

    Barbara Schamberger, Ricardo Ziege, Karine Anselme, Martine Ben Amar, Michał Bykowski, André PG Castro, Amaia Cipitria, Rhoslyn A Coles, Rumiana Dimova, Michaela Eder, et al. Curvature in biological systems: its quantification, emer- gence, and implications across the scales.advanced materials, 35(13):2206110,

    work page

  37. [37]

    A. M. Turing. The chemical basis of morphogensis.Phil. Trans. Roy. Soc., B (237):37–72, 1952. 2

    work page 1952

  38. [38]

    F ARIV AR Frank Jülicher

    David Zwicker, Rabea Seyboldt, Christoph A Weber, Anthony A Hyman, and 28 F. F ARIV AR Frank Jülicher. Growth and division of active droplets provides a model for pro- tocells.Nature Physics, 13(4):408–413, 2017. 4, 5 1 Institute of Atmospheric Sciences and Climate (CNR-ISAC), Rome, Italy Email address:faezehfarivar@cnr.it

    work page 2017