Anomalous Diffusion as Structural Memory: An Extended Structural Dynamics Approach

Patrick BarAvi

arxiv: 2605.16337 · v1 · pith:YGKLZMG4new · submitted 2026-05-07 · ❄️ cond-mat.stat-mech · cond-mat.soft· physics.bio-ph

Anomalous Diffusion as Structural Memory: An Extended Structural Dynamics Approach

Patrick BarAvi This is my paper

Pith reviewed 2026-05-20 23:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.softphysics.bio-ph

keywords anomalous diffusionsubdiffusionstructural dynamicsmemory kernelphase space projectionbiological macromoleculescrowded mediainternal modes

0 comments

The pith

Subdiffusion in biology arises when dynamics of structured molecules are projected onto translation alone.

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

The paper argues that subdiffusion appears anomalous only because standard models treat molecules as point particles with an incomplete phase space. In the Extended Structural Dynamics framework each molecule carries translational, orientational, and deformational degrees of freedom. When the full dynamics are projected down to the translational coordinates, a memory kernel is generated automatically by the projection step itself. The subdiffusion exponent is then fixed by the spectrum of internal modes, which can be read out independently from B-factors, NMR order parameters, or molecular dynamics runs without any fit to the transport data. The approach yields four concrete, testable predictions and matches existing measurements on proteins in crowded solutions using only those independent structural inputs.

Core claim

When dynamics on the full extended phase space of a structured particle are projected onto the translational subspace alone, a memory kernel emerges from the projection without phenomenological postulate. The subdiffusion exponent is determined by the internal mode spectrum, independently measurable from B-factors, NMR order parameters, or molecular dynamics simulations, without fitting to transport data.

What carries the argument

The projection operation from the complete extended phase space (translational plus orientational plus deformational modes) onto the translational coordinates alone.

If this is right

Subdiffusion strength correlates with molecular flexibility because more internal modes strengthen the emergent memory.
Temperature drives a crossover to normal diffusion once thermal energy exceeds the characteristic scale set by internal mode frequencies.
A non-zero rotation-translation cross-correlation spectrum appears and encodes internal dynamics, identically zero in point-particle models.
Memory timescales scale as the square of particle size.

Where Pith is reading between the lines

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

The same projection logic may generate apparent anomalies in any system whose particles possess internal structure, such as polymers or colloids.
Varying particle size while holding internal-mode frequencies fixed would provide a clean test of the predicted quadratic scaling of memory time.
Non-Gaussian displacement statistics observed in crowded media could receive a similar structural-memory explanation once the full phase space is restored.

Load-bearing premise

The internal mode spectrum and other structural parameters can be obtained independently of the transport measurements themselves.

What would settle it

A direct test in which the subdiffusion exponent calculated from an independently measured internal-mode spectrum (via B-factors or NMR) fails to match the observed exponent in the same system.

read the original abstract

Sub-diffusion in biological systems is conventionally treated as anomalous, requiring fractional derivatives, heavy-tailed waiting times, or fitted memory kernels. We argue that this anomaly is an artifact of an incomplete phase space. Standard frameworks model diffusing particles as points. Biological molecules are not points. They are three-dimensional deformable entities whose position, orientation, and internal structure are irreducible physical properties, not modeling conveniences appended to a point mass. Within the Extended Structural Dynamics (ESD) framework, each particle is a primitive structured entity with translational, orientational, and deformational degrees of freedom. When dynamics on this full phase space are projected onto the translational subspace alone, a memory kernel emerges from the projection without phenomenological postulate. The subdiffusion exponent is determined by the internal mode spectrum, independently measurable from B-factors, NMR order parameters, or molecular dynamics simulations, without fitting to transport data. Four falsifiable predictions follow: subdiffusion strength correlates with molecular flexibility; temperature drives crossover to normal diffusion at a characteristic energy scale set by internal mode frequencies; a non-zero rotation-translation cross-correlation spectrum encodes internal dynamics, identically zero in point-particle models; and memory timescales scale as the square of particle size. Quantitative consistency with experimental observations for proteins in crowded media is demonstrated using independently estimated structural parameters. What appears anomalous from the point-particle perspective is the expected behavior of structured matter projected onto an impoverished description. The anomaly is not in the physics. It is in the phase space.

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.

Desk Editor's Note private letter to a colleague

This paper frames subdiffusion as a projection effect from a molecule's full structural phase space, but the memory kernel still appears to require dynamical couplings not fixed solely by B-factors or NMR data.

read the letter

The main thing here is that the work treats subdiffusion as what you get when you project the translational, orientational, and deformational motions of a real molecule down to just its center-of-mass position. The claim is that a memory kernel appears automatically from that projection, and the subdiffusion exponent is set by the internal mode spectrum that can be read off from B-factors, NMR order parameters, or simulations without fitting to the diffusion data itself. Four concrete predictions are listed: correlation with molecular flexibility, a temperature-driven crossover to normal diffusion, nonzero rotation-translation cross-correlations, and memory times scaling with particle size squared. Quantitative checks against protein data in crowded media are said to use independently estimated structural parameters. That framing is the actual novelty; it moves away from adding fractional derivatives or waiting-time distributions by hand and instead ties the anomaly to an incomplete phase space. The predictions are specific enough that they could be tested directly, which is a strength. The paper does well in spelling out why point-particle models miss the internal degrees of freedom that biological molecules actually have. The soft spot is the one the stress-test note flags. B-factors and NMR parameters give equilibrium variances and order, but the memory kernel in any projection-operator approach is built from the time correlations of the fluctuating forces orthogonal to the chosen subspace. Those correlations depend on the off-diagonal couplings between translational velocity and the internal modes plus the relaxation rates of those modes. The abstract does not show whether the ESD construction supplies an explicit, structure-only rule for those couplings or whether some modeling choices remain. If the full derivation still leaves the exponent sensitive to how the couplings are set, then the independence from transport data is weaker than stated. This is aimed at biophysicists and soft-matter people who already work with anomalous diffusion in crowded or cellular environments and want a mechanism grounded in measurable molecular properties. A reader looking for alternatives to phenomenological kernels would find the predictions and the structural starting point worth examining. It deserves peer review so the projection steps and any quantitative comparisons can be checked by people familiar with both molecular dynamics and projection formalisms.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces the Extended Structural Dynamics (ESD) framework, treating biological molecules as structured entities with translational, orientational, and deformational degrees of freedom rather than point particles. It argues that projecting the full phase-space dynamics onto the translational subspace alone generates a memory kernel without phenomenological input, such that the subdiffusion exponent is fixed by the internal mode spectrum (measurable independently from B-factors, NMR order parameters, or MD simulations). Four falsifiable predictions are stated, and quantitative consistency with protein diffusion in crowded media is claimed using independently estimated structural parameters.

Significance. If the projection step yields a memory kernel whose long-time decay is demonstrably fixed by equilibrium structural observables without additional fitted couplings, the work would supply a structural origin for subdiffusion that unifies transport anomalies with independently measurable molecular properties. The emphasis on falsifiable predictions and the avoidance of fractional derivatives or ad-hoc kernels are strengths that could influence modeling of intracellular transport.

major comments (1)

[Abstract and ESD projection derivation] The central claim (abstract) that the subdiffusion exponent is completely determined by the internal mode spectrum rests on the projection operator producing a memory kernel whose form is fixed solely by equilibrium structural data. In any projection formalism the memory kernel is the autocorrelation of the fluctuating force orthogonal to the translational subspace; this autocorrelation is shaped by the dynamical coupling matrix between translational velocity and internal coordinates. B-factors and NMR order parameters supply equilibrium variances but do not uniquely fix the off-diagonal coupling strengths or relaxation rates. The manuscript must supply the explicit, parameter-free rule (e.g., from a microscopic Hamiltonian or structural constraint) that determines these couplings; absent that rule the exponent remains sensitive to modeling choices that are not transport-independent.

minor comments (2)

[Abstract] The abstract asserts 'quantitative consistency with experimental observations' using independently estimated parameters but does not identify the specific datasets, the structural parameters employed, or the figure/table that displays the comparison; a brief pointer would improve traceability.
[Predictions paragraph] The four falsifiable predictions are enumerated but their explicit functional forms (e.g., the expected scaling of memory time with particle size squared) are not given; adding one or two quantitative statements would make the predictions easier to test.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We have carefully considered the major comment and provide our response below. We believe the concerns can be addressed through clarification in the revised version.

read point-by-point responses

Referee: The central claim (abstract) that the subdiffusion exponent is completely determined by the internal mode spectrum rests on the projection operator producing a memory kernel whose form is fixed solely by equilibrium structural data. In any projection formalism the memory kernel is the autocorrelation of the fluctuating force orthogonal to the translational subspace; this autocorrelation is shaped by the dynamical coupling matrix between translational velocity and internal coordinates. B-factors and NMR order parameters supply equilibrium variances but do not uniquely fix the off-diagonal coupling strengths or relaxation rates. The manuscript must supply the explicit, parameter-free rule (e.g., from a microscopic Hamiltonian or structural constraint) that determines these couplings; absent that rule the exponent remains sensitive to modeling choices that are not transport-independent.

Authors: We agree that clarifying the determination of the coupling matrix is important. In the ESD approach, the full phase space includes translational, rotational, and deformational coordinates, with the equilibrium distribution given by the structural data (B-factors provide the variances of internal modes). The projection onto the translational subspace uses the Mori-Zwanzig formalism, where the memory kernel is indeed the autocorrelation of the orthogonal force. The dynamical couplings are fixed by the microscopic model: we model the molecule as a rigid body with internal harmonic modes whose frequencies and eigenvectors are determined from the Hessian of the potential energy, constrained by the molecular structure. These are obtained independently from MD simulations or normal mode analysis, without reference to diffusion data. The relaxation rates follow from the mode frequencies and damping. Thus, the subdiffusion exponent, which depends on the spectrum of these modes, is fixed by structural observables. We will revise the manuscript to include an explicit derivation of the coupling matrix from the structural Hamiltonian in the Methods section to make this parameter-free aspect clearer. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as projection from independent structural inputs

full rationale

The paper asserts that projecting the extended structural dynamics phase space (translation + orientation + deformation) onto the translational subspace yields a memory kernel whose long-time behavior sets the subdiffusion exponent via the internal mode spectrum. This spectrum is claimed to be obtained from B-factors, NMR order parameters, or MD simulations without reference to transport data. No equations, self-citations, or fitted parameters are exhibited in the provided text that reduce the exponent or kernel to a quantity defined in terms of the diffusion measurements themselves. The central claim therefore remains self-contained against external structural benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review is based on the abstract alone; the ledger therefore reflects only claims stated there. The memory kernel is asserted to emerge from projection, implying no free parameter is introduced for the kernel itself, yet the internal mode spectrum remains the load-bearing input.

axioms (1)

domain assumption Biological molecules are three-dimensional deformable entities whose position, orientation, and internal structure are irreducible physical properties.
Stated explicitly as the starting point of the ESD framework in the abstract.

invented entities (1)

Extended Structural Dynamics (ESD) framework no independent evidence
purpose: To treat each particle as a primitive structured entity with translational, orientational, and deformational degrees of freedom whose projection yields memory.
New modeling framework introduced in the paper to derive the memory kernel.

pith-pipeline@v0.9.0 · 5794 in / 1486 out tokens · 81326 ms · 2026-05-20T23:54:09.971904+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

When dynamics on this full phase space are projected onto the translational subspace alone, a memory kernel emerges from the projection without phenomenological postulate. ... Γ(t) = λ²/m ∑ sin(ω_α t)/ω_α
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the subdiffusion exponent α is determined by the internal mode spectrum ... ρ(ω) = ρ0 ω^β, β ≈ 0.8–1.2

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

53 extracted references · 53 canonical work pages

[1]

Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen

Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322, 549–560 (1905)

work page 1905
[2]

Mouvement brownien et réalité moléculaire

Perrin, J. Mouvement brownien et réalité moléculaire. Ann. Chim. Phys. 18, 5–114 (1909)

work page 1909
[3]

Extended Structural Dynamics: Emergent irreversibility from reversible dynamics

BarAvi, P. Extended Structural Dynamics: Emergent irreversibility from reversible dynamics. arXiv:2505.09650 [physics.class-ph] (2025)

work page arXiv 2025
[4]

Structural viscosity, thermal waves, and the Mpemba effect from Extended Structural Dynamics

BarAvi, P. Structural viscosity, thermal waves, and the Mpemba effect from Extended Structural Dynamics. arXiv:2603.02249 [physics.class-ph] (2026a)

work page arXiv
[5]

Extended Structural Dynamics and the Lorentz–Abraham–Dirac equation: A deformable charge interpretation

BarAvi, P. Extended Structural Dynamics and the Lorentz–Abraham–Dirac equation: A deformable charge interpretation. arXiv:2603.11064 [physics.class-ph] (2026b). [5a] BarAvi, P. (2026). Extended Structural Dynamics of Charged Media: Kinetic Theory, Generalized Plasma. Transport Phenomena, De Gruyter Brill, in press

work page arXiv 2026
[6]

& Nilsson, T

Weiss, M., Elsner, M., Kartberg, F. & Nilsson, T. Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. Biophys. J. 87, 3518–3524 (2004)

work page 2004
[7]

Banks, D. S. & Fradin, C. Anomalous diffusion of proteins due to molecular crowding. Biophys. J. 89, 2960–2971 (2005)

work page 2005
[8]

& Cox, E

Golding, I. & Cox, E. C. Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96, 098102 (2006)

work page 2006
[9]

M.-S., Javanainen, M

Jeon, J.-H., Monne, H. M.-S., Javanainen, M. & Metzler, R. Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. Phys. Rev. Lett. 109, 188103 (2012)

work page 2012
[10]

Jeon, J.-H. et al. In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011)

work page 2011
[11]

C., Spakowitz, A

Weber, S. C., Spakowitz, A. J. & Theriot, J. A. Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm. Phys. Rev. Lett. 104, 238102 (2010)

work page 2010
[12]

Crater, J. S. & Carrier, R. L. Barrier properties of gastrointestinal mucus to nanoparticle transport. Macromol. Biosci. 10, 1473–1483 (2010)

work page 2010
[13]

Manzo, C. et al. Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity. Phys. Rev. X 5, 011021 (2015)

work page 2015
[14]

& Franosch, T

Höfling, F. & Franosch, T. Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 76, 046602 (2013). 34

work page 2013
[15]

Dix, J. A. & Verkman, A. S. Crowding effects on diffusion in solutions and cells. Annu. Rev. Biophys. 37, 247–263 (2008)

work page 2008
[16]

& Klafter, J

Metzler, R. & Klafter, J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

work page 2000
[17]

Fractional Calculus and Waves in Linear Viscoelasticity

Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press (2010)

work page 2010
[18]

Montroll, E. W. & Weiss, G. H. Random walks on lattices. II. J. Math. Phys. 6, 167–181 (1965)

work page 1965
[19]

Weak ergodicity breaking and aging in disordered systems

Bouchaud, J.-P. Weak ergodicity breaking and aging in disordered systems. J. Phys. I France 2, 1705–1713 (1992)

work page 1992
[20]

The fluctuation-dissipation theorem

Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966)

work page 1966
[21]

Viscoelastic subdiffusion: Generalized Langevin equation approach

Goychuk, I. Viscoelastic subdiffusion: Generalized Langevin equation approach. Adv. Chem. Phys. 150, 187–253 (2012)

work page 2012
[22]

& Hynes, J

Laage, D., Elsaesser, T. & Hynes, J. T. Water dynamics in the hydration shells of biomolecules. Chem. Rev. 117, 10694–10725 (2017)

work page 2017
[23]

Water as an active constituent in cell biology

Ball, P. Water as an active constituent in cell biology. Chem. Rev. 108, 74–108 (2008)

work page 2008
[24]

& Kern, D

Henzler-Wildman, K. & Kern, D. Dynamic personalities of proteins. Nature 450, 964–972 (2007)

work page 2007
[25]

Frauenfelder, H., Sligar, S. G. & Wolynes, P. G. The energy landscapes and motions of proteins. Science 254, 1598–1603 (1991)

work page 1991
[26]

Teilum, K., Olsen, J. G. & Kragelund, B. B. Functional aspects of protein flexibility. Cell. Mol. Life Sci. 66, 2231–2247 (2009)

work page 2009
[27]

Nonequilibrium Statistical Mechanics

Zwanzig, R. Nonequilibrium Statistical Mechanics. Oxford University Press (2001)

work page 2001
[28]

Projection Operator Techniques in Nonequilibrium Statistical Mechanics

Grabert, H. Projection Operator Techniques in Nonequilibrium Statistical Mechanics. Springer (1982)

work page 1982
[29]

& Bahar, I

Cui, Q. & Bahar, I. Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems. Chapman & Hall/CRC (2006)

work page 2006
[30]

& Stern, P

Levitt, M., Sander, C. & Stern, P. S. Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme. J. Mol. Biol. 181, 423–447 (1985)

work page 1985
[31]

& Sanejouand, Y .-H

Tama, F. & Sanejouand, Y .-H. Conformational change of proteins arising from normal mode calculations. Protein Eng. 14, 1–6 (2001). 35

work page 2001
[32]

& Argos, P

Carugo, O. & Argos, P. Correlation between side chain mobility and conformation in protein structures. Protein Eng. 10, 777–787 (1997)

work page 1997
[33]

F., Dewan, J

Tilton, R. F., Dewan, J. C. & Petsko, G. A. Effects of temperature on protein structure and dynamics. Biochemistry 31, 2469–2481 (1992)

work page 1992
[34]

Palmer, A. G. NMR characterization of the dynamics of biomacromolecules. Chem. Rev. 104, 3623–3640 (2004)

work page 2004
[35]

Jarymowycz, V . A. & Stone, M. J. Fast time scale dynamics of protein backbones: NMR relaxation methods. Chem. Rev. 106, 1624–1671 (2006)

work page 2006
[36]

& Brenner, H

Happel, J. & Brenner, H. Low Reynolds Number Hydrodynamics. Martinus Nijhoff (1983)

work page 1983
[37]

Ohmachi, M. et al. Fluorescence microscopy for simultaneous observation of 3D orientation and movement and its application to quantum rod-tagged myosin V . Proc. Natl. Acad. Sci. USA 109, 5294–5298 (2012)

work page 2012
[38]

A., Quinlan, M

Rosenberg, S. A., Quinlan, M. E., Forkey, J. N. & Goldman, Y . E. Rotational motions of macromolecules by single-molecule fluorescence microscopy. Acc. Chem. Res. 38, 583–593 (2005)

work page 2005
[39]

Metzler, R., Jeon, J.-H., Cherstvy, A. G. & Barkai, E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164 (2014)

work page 2014
[40]

& Weiss, M

Szymanski, J. & Weiss, M. Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. 103, 038102 (2009)

work page 2009
[41]

& Feig, M

Nawrocki, G., Wang, P.-H., Yu, I., Sugita, Y . & Feig, M. Slow-down in diffusion in crowded protein solutions correlates with transient cluster formation. J. Phys. Chem. B 121, 11072–11084 (2017)

work page 2017
[42]

Radmacher, M., Tillmann, R. W. & Gaub, H. E. Imaging viscoelasticity by force modulation with the atomic force microscope. Biophys. J. 64, 735–742 (1993)

work page 1993
[43]

Svergun, D. I. et al. Protein hydration in solution: experimental observation by x-ray and neutron scattering. Proc. Natl. Acad. Sci. USA 95, 2267–2272 (1998)

work page 1998
[44]

& Onuchic, J

Levy, Y . & Onuchic, J. N. Water mediation in protein folding and molecular recognition. Annu. Rev. Biophys. Biomol. Struct. 35, 389–415 (2006)

work page 2006
[45]

Manton, J. D. et al. Concepts for high-speed multiplane confocal microscopy. J. Microsc. 281, 157–169 (2021)

work page 2021
[46]

Gwosch, K. C. et al. MINFLUX nanoscopy delivers 3D multicolor nanometer resolution in cells. Nat. Methods 17, 217–224 (2020). 36

work page 2020
[47]

J., Woolf, T

Michaud-Agrawal, N., Denning, E. J., Woolf, T. B. & Beckstein, O. MDAnalysis: a toolkit for the analysis of molecular dynamics simulations. J. Comput. Chem. 32, 2319–2327 (2011)

work page 2011
[48]

Shaw, D. E. et al. Anton 2: raising the bar for performance and programmability in a special-purpose molecular dynamics supercomputer. Proc. IEEE Int. Conf. High Perform. Comput. Netw. Storage Anal. 41, 41–53 (2014)

work page 2014
[49]

& Metzler, R

Barkai, E., Garini, Y . & Metzler, R. Strange kinetics of single molecules in living cells. Phys. Today 65, 29–35 (2012)

work page 2012
[50]

& Weiss, M

Sabri, A., Xu, X., Krapf, D. & Weiss, M. Elucidating the origin of heterogeneous anomalous diffusion in the cytoplasm of mammalian cells. Nat. Commun. 13, 6717 (2022). Appendix A: Derivation of the Memory Kernel from Extended Structural Dynamics This appendix summarizes the relevant results from the ESD hydrodynamic framework [4]. The reader is referred t...

work page 2022
[51]

Scale separation: 𝜏int ≪ 𝜏obs, where 𝜏int ∼ 𝜔𝛼 −1 is the fastest internal relaxation time and 𝜏obs is the observation time

work page
[52]

Weak coupling: 𝜆 ∣ ∇𝑉ext ∣≪ 𝑘𝐵𝑇/𝑎, where 𝑎 is the particle size

work page
[53]

38 Appendix B: Mode Density and Power-Law Spectrum in Proteins This appendix provides background on the mode density of proteins and the power-law approximation used in Section 2.5

Linear response: Internal deformations remain small compared to particle dimensions For the physiological conditions considered in this paper (proteins in cytoplasm at 𝑇 = 300 K), these conditions are satisfied. 38 Appendix B: Mode Density and Power-Law Spectrum in Proteins This appendix provides background on the mode density of proteins and the power-la...

work page

[1] [1]

Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen

Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322, 549–560 (1905)

work page 1905

[2] [2]

Mouvement brownien et réalité moléculaire

Perrin, J. Mouvement brownien et réalité moléculaire. Ann. Chim. Phys. 18, 5–114 (1909)

work page 1909

[3] [3]

Extended Structural Dynamics: Emergent irreversibility from reversible dynamics

BarAvi, P. Extended Structural Dynamics: Emergent irreversibility from reversible dynamics. arXiv:2505.09650 [physics.class-ph] (2025)

work page arXiv 2025

[4] [4]

Structural viscosity, thermal waves, and the Mpemba effect from Extended Structural Dynamics

BarAvi, P. Structural viscosity, thermal waves, and the Mpemba effect from Extended Structural Dynamics. arXiv:2603.02249 [physics.class-ph] (2026a)

work page arXiv

[5] [5]

Extended Structural Dynamics and the Lorentz–Abraham–Dirac equation: A deformable charge interpretation

BarAvi, P. Extended Structural Dynamics and the Lorentz–Abraham–Dirac equation: A deformable charge interpretation. arXiv:2603.11064 [physics.class-ph] (2026b). [5a] BarAvi, P. (2026). Extended Structural Dynamics of Charged Media: Kinetic Theory, Generalized Plasma. Transport Phenomena, De Gruyter Brill, in press

work page arXiv 2026

[6] [6]

& Nilsson, T

Weiss, M., Elsner, M., Kartberg, F. & Nilsson, T. Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. Biophys. J. 87, 3518–3524 (2004)

work page 2004

[7] [7]

Banks, D. S. & Fradin, C. Anomalous diffusion of proteins due to molecular crowding. Biophys. J. 89, 2960–2971 (2005)

work page 2005

[8] [8]

& Cox, E

Golding, I. & Cox, E. C. Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96, 098102 (2006)

work page 2006

[9] [9]

M.-S., Javanainen, M

Jeon, J.-H., Monne, H. M.-S., Javanainen, M. & Metzler, R. Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. Phys. Rev. Lett. 109, 188103 (2012)

work page 2012

[10] [10]

Jeon, J.-H. et al. In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011)

work page 2011

[11] [11]

C., Spakowitz, A

Weber, S. C., Spakowitz, A. J. & Theriot, J. A. Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm. Phys. Rev. Lett. 104, 238102 (2010)

work page 2010

[12] [12]

Crater, J. S. & Carrier, R. L. Barrier properties of gastrointestinal mucus to nanoparticle transport. Macromol. Biosci. 10, 1473–1483 (2010)

work page 2010

[13] [13]

Manzo, C. et al. Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity. Phys. Rev. X 5, 011021 (2015)

work page 2015

[14] [14]

& Franosch, T

Höfling, F. & Franosch, T. Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 76, 046602 (2013). 34

work page 2013

[15] [15]

Dix, J. A. & Verkman, A. S. Crowding effects on diffusion in solutions and cells. Annu. Rev. Biophys. 37, 247–263 (2008)

work page 2008

[16] [16]

& Klafter, J

Metzler, R. & Klafter, J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

work page 2000

[17] [17]

Fractional Calculus and Waves in Linear Viscoelasticity

Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press (2010)

work page 2010

[18] [18]

Montroll, E. W. & Weiss, G. H. Random walks on lattices. II. J. Math. Phys. 6, 167–181 (1965)

work page 1965

[19] [19]

Weak ergodicity breaking and aging in disordered systems

Bouchaud, J.-P. Weak ergodicity breaking and aging in disordered systems. J. Phys. I France 2, 1705–1713 (1992)

work page 1992

[20] [20]

The fluctuation-dissipation theorem

Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966)

work page 1966

[21] [21]

Viscoelastic subdiffusion: Generalized Langevin equation approach

Goychuk, I. Viscoelastic subdiffusion: Generalized Langevin equation approach. Adv. Chem. Phys. 150, 187–253 (2012)

work page 2012

[22] [22]

& Hynes, J

Laage, D., Elsaesser, T. & Hynes, J. T. Water dynamics in the hydration shells of biomolecules. Chem. Rev. 117, 10694–10725 (2017)

work page 2017

[23] [23]

Water as an active constituent in cell biology

Ball, P. Water as an active constituent in cell biology. Chem. Rev. 108, 74–108 (2008)

work page 2008

[24] [24]

& Kern, D

Henzler-Wildman, K. & Kern, D. Dynamic personalities of proteins. Nature 450, 964–972 (2007)

work page 2007

[25] [25]

Frauenfelder, H., Sligar, S. G. & Wolynes, P. G. The energy landscapes and motions of proteins. Science 254, 1598–1603 (1991)

work page 1991

[26] [26]

Teilum, K., Olsen, J. G. & Kragelund, B. B. Functional aspects of protein flexibility. Cell. Mol. Life Sci. 66, 2231–2247 (2009)

work page 2009

[27] [27]

Nonequilibrium Statistical Mechanics

Zwanzig, R. Nonequilibrium Statistical Mechanics. Oxford University Press (2001)

work page 2001

[28] [28]

Projection Operator Techniques in Nonequilibrium Statistical Mechanics

Grabert, H. Projection Operator Techniques in Nonequilibrium Statistical Mechanics. Springer (1982)

work page 1982

[29] [29]

& Bahar, I

Cui, Q. & Bahar, I. Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems. Chapman & Hall/CRC (2006)

work page 2006

[30] [30]

& Stern, P

Levitt, M., Sander, C. & Stern, P. S. Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme. J. Mol. Biol. 181, 423–447 (1985)

work page 1985

[31] [31]

& Sanejouand, Y .-H

Tama, F. & Sanejouand, Y .-H. Conformational change of proteins arising from normal mode calculations. Protein Eng. 14, 1–6 (2001). 35

work page 2001

[32] [32]

& Argos, P

Carugo, O. & Argos, P. Correlation between side chain mobility and conformation in protein structures. Protein Eng. 10, 777–787 (1997)

work page 1997

[33] [33]

F., Dewan, J

Tilton, R. F., Dewan, J. C. & Petsko, G. A. Effects of temperature on protein structure and dynamics. Biochemistry 31, 2469–2481 (1992)

work page 1992

[34] [34]

Palmer, A. G. NMR characterization of the dynamics of biomacromolecules. Chem. Rev. 104, 3623–3640 (2004)

work page 2004

[35] [35]

Jarymowycz, V . A. & Stone, M. J. Fast time scale dynamics of protein backbones: NMR relaxation methods. Chem. Rev. 106, 1624–1671 (2006)

work page 2006

[36] [36]

& Brenner, H

Happel, J. & Brenner, H. Low Reynolds Number Hydrodynamics. Martinus Nijhoff (1983)

work page 1983

[37] [37]

Ohmachi, M. et al. Fluorescence microscopy for simultaneous observation of 3D orientation and movement and its application to quantum rod-tagged myosin V . Proc. Natl. Acad. Sci. USA 109, 5294–5298 (2012)

work page 2012

[38] [38]

A., Quinlan, M

Rosenberg, S. A., Quinlan, M. E., Forkey, J. N. & Goldman, Y . E. Rotational motions of macromolecules by single-molecule fluorescence microscopy. Acc. Chem. Res. 38, 583–593 (2005)

work page 2005

[39] [39]

Metzler, R., Jeon, J.-H., Cherstvy, A. G. & Barkai, E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164 (2014)

work page 2014

[40] [40]

& Weiss, M

Szymanski, J. & Weiss, M. Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. 103, 038102 (2009)

work page 2009

[41] [41]

& Feig, M

Nawrocki, G., Wang, P.-H., Yu, I., Sugita, Y . & Feig, M. Slow-down in diffusion in crowded protein solutions correlates with transient cluster formation. J. Phys. Chem. B 121, 11072–11084 (2017)

work page 2017

[42] [42]

Radmacher, M., Tillmann, R. W. & Gaub, H. E. Imaging viscoelasticity by force modulation with the atomic force microscope. Biophys. J. 64, 735–742 (1993)

work page 1993

[43] [43]

Svergun, D. I. et al. Protein hydration in solution: experimental observation by x-ray and neutron scattering. Proc. Natl. Acad. Sci. USA 95, 2267–2272 (1998)

work page 1998

[44] [44]

& Onuchic, J

Levy, Y . & Onuchic, J. N. Water mediation in protein folding and molecular recognition. Annu. Rev. Biophys. Biomol. Struct. 35, 389–415 (2006)

work page 2006

[45] [45]

Manton, J. D. et al. Concepts for high-speed multiplane confocal microscopy. J. Microsc. 281, 157–169 (2021)

work page 2021

[46] [46]

Gwosch, K. C. et al. MINFLUX nanoscopy delivers 3D multicolor nanometer resolution in cells. Nat. Methods 17, 217–224 (2020). 36

work page 2020

[47] [47]

J., Woolf, T

Michaud-Agrawal, N., Denning, E. J., Woolf, T. B. & Beckstein, O. MDAnalysis: a toolkit for the analysis of molecular dynamics simulations. J. Comput. Chem. 32, 2319–2327 (2011)

work page 2011

[48] [48]

Shaw, D. E. et al. Anton 2: raising the bar for performance and programmability in a special-purpose molecular dynamics supercomputer. Proc. IEEE Int. Conf. High Perform. Comput. Netw. Storage Anal. 41, 41–53 (2014)

work page 2014

[49] [49]

& Metzler, R

Barkai, E., Garini, Y . & Metzler, R. Strange kinetics of single molecules in living cells. Phys. Today 65, 29–35 (2012)

work page 2012

[50] [50]

& Weiss, M

Sabri, A., Xu, X., Krapf, D. & Weiss, M. Elucidating the origin of heterogeneous anomalous diffusion in the cytoplasm of mammalian cells. Nat. Commun. 13, 6717 (2022). Appendix A: Derivation of the Memory Kernel from Extended Structural Dynamics This appendix summarizes the relevant results from the ESD hydrodynamic framework [4]. The reader is referred t...

work page 2022

[51] [51]

Scale separation: 𝜏int ≪ 𝜏obs, where 𝜏int ∼ 𝜔𝛼 −1 is the fastest internal relaxation time and 𝜏obs is the observation time

work page

[52] [52]

Weak coupling: 𝜆 ∣ ∇𝑉ext ∣≪ 𝑘𝐵𝑇/𝑎, where 𝑎 is the particle size

work page

[53] [53]

38 Appendix B: Mode Density and Power-Law Spectrum in Proteins This appendix provides background on the mode density of proteins and the power-law approximation used in Section 2.5

Linear response: Internal deformations remain small compared to particle dimensions For the physiological conditions considered in this paper (proteins in cytoplasm at 𝑇 = 300 K), these conditions are satisfied. 38 Appendix B: Mode Density and Power-Law Spectrum in Proteins This appendix provides background on the mode density of proteins and the power-la...

work page