Quantum Information Flow in Microtubule Tryptophan Networks

The gist The results reveal a strong dependence of both the direction and persistence of information flow on the type of initial preparation, identifying structural and dynamical conditions that transiently preserve nonclassical correlations in microtubules.

Model Framework

The study employs a Lindblad master equation to track how correlations are generated, routed, and dissipated by incorporating explicit site geometries and dipole orientations into an effective non-Hermitian Hamiltonian (Page 2). This approach allows for a physically consistent open-system description where the dynamics are governed by the equation dρ(t)/dt = -i[H, ρ(t)] + Σ j L jρ(t)L j† - 1/2 L j†L j, ρ(t) (Page 3). The Hamiltonian H is constructed from the on-site excitation energies H0 and the dipole-dipole coupling term ∆, which depends on site positions and transition dipole moments (Page 3).

Quantifying Information Flow

The researchers quantify the emerging quantum-informational structure using several metrics to evaluate correlation persistence (Page 2). These metrics include:

1. The L1 norm of coherence, which monitors the information carried by coherent excitation delocalization (Page 2).

2. Correlated coherence, a basis-independent measure capturing how much information is shared between microtubule substructures (Page 2).

3. Logarithmic negativity, used as an entanglement witness to quantify entanglement within and between selected chromophore sub-networks (Page 2).

Initial State Dependence and Dynamics

The analysis compares five distinct initial preparations to probe different routes for correlation propagation (Page 3):

(i) superradiant eigenstates of Heff:

When initialized in the superradiant eigenstate, the excitation undergoes rapid collective dissipation, with site populations decaying almost synchronously, reflecting strong collective coupling to the radiation field (Page 6). The L1 norm of coherence drops sharply within about 1 ns, indicating that information flow is dominated by rapid leakage rather than internal exchange (Page 6).

(ii) subradiant eigenstates of Heff:

Initializing in the most subradiant eigenstate yields a much slower release of excitation, with populations remaining within the network for tens of nanoseconds, indicating strong suppression of radiative loss (Page 6). In this setting, the dominant direction is internal rather than outward, with excitations and correlations circulating among chromophores before any leakage to the environment (Page 6).

(iii) a maximally coherent single excitation:

A fully coherent delocalized state shows that correlations propagate through the network as oscillatory information flow between sites, with the L1 norm remaining high over the full 0–15 ns window, confirming that internal information exchange persists (Page 7).

(iv) a fully incoherent mixed state:

For a maximally mixed initial state, the absence of initial coherence prevents any internal information circulation, and populations decay monotonically and nearly uniformly across all sites (Page 8). This highlights that coherence is a prerequisite for sustained correlation transport and retention of quantum information within the microtubule network (Page 8).

Structural and Environmental Effects

The study systematically examines how geometry and disorder modulate correlation dynamics (Page 3):

1. Embedding single tubulin units into larger dimers and spirals reshapes pairwise correlation maps, enabling site-selective routing (Page 3).

2. Scaling to larger ordered lattices strengthens both export and retention channels, whereas static energetic and structural disorder suppresses long-range transport and reduces overall correlation transfer (Page 3).

3. Static or structural disorder—either through random diagonal site-energy fluctuations or MD-sampled geometric variability—substantially disrupts long-range coherence in the superradiant state, localizing it and fragmenting it (Page 11).

Timescales and Memory Effects

The results establish two distinct timescales for information flow (Page 17):

(ps–ns scale):

Coherent redistribution within a local Trp network occurs on the ps–ns scale set by dipole couplings, showing oscillatory exchange with surrounding tubulins in larger structures (Page 17).

(ms scale):

The radiative lifetimes of collective eigenmodes can extend to much longer scales in ordered assemblies, while disorder tends to localize bright components and compress lifetime contrasts (Page 16). Furthermore, non-Markovian backflow is detected in reduced tubulin subsystems, indicating that nearby excitation events in a finite spiral need not behave as independent memoryless decays (Page 23).

Conclusion

The findings identify microtubule tryptophan networks as structured reservoirs where symmetry, spatial arrangement, and initial phase determine whether information is rapidly broadcast through bright channels or preferentially stored in dark subspaces (Page 17). The combination of non-Hermitian eigenmode structure with quantum information metrics provides a versatile toolbox for biological systems to clarify when coherence is delocalized, when it is protected, and when memory effects reshape the local flow of information (Page 18). This framework guides experiments aimed at testing whether optical excitations couple measurably to downstream microtubule-associated dynamics (Page 17).

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AUTHOR CONTRIBUTIONS Conceptualization: L.G., O.P., and T.J.A.C.; Methodology: L.G., O.P., and T.J.A.C.; Formal analysis: L.G.; Resources: T.J.A.C.; Data curation: L.G.; Writing—original draft preparation: L.G.; Writing—review and editing: L.G., O.P., and T.J A C.; Visualization: L.G.; Supervision: T.J A C. and O P.; Project administration: T J A C. Funding acquisition: T J A C. The research was undertaken in part thanks to funding to T J A C from the Canada Research Chairs Program (CRC-2022-00204) and the University of Waterloo (Page 18). Data supporting the findings of this study are available from the corresponding author (L.G.) upon reasonable request (Page 18). The authors used an AI-assisted language tool to help improve the clarity and readability of the text, but all AI-assisted content was reviewed and edited by the authors, who take full responsibility for the integrity and accuracy of the final manuscript (Page 18). This research was enabled in part by support provided by the Digital Research Alliance of Canada (alliancecan.ca) to T.J A C. (Page 18). The paper is organized as follows: Section II describes the model and methods; Section III presents the results; and Section IV discusses the findings and summarizes the study’s implications (Page 2).

Appendix A Construction of Ordered and Disordered Microtubule Geometries: Ordered microtubules are assembled by repeating the 1JFF tubulin dimer and arranging dimers with a left-handed spiral geometry following prior work [10, 12] and using the code provided by Patwa et al. [11] (Page 26). Geometric variability is sampled by molecular dynamics of a tubulin dimer and by assembling microtubules from snapshot structures (Page 26).

Appendix B Connection Between the Non-Hermitian Hamiltonian and the Lindblad Master Equation: We define the effective non-Hermitian Hamiltonian as Heff = H - i/2 Σ µ γ µ L µ† L µ (Page 20). The full master equation then reads dρ(t)/dt = -i[H, ρ(t)] + Σ j L jρ(t)L j† - 1/2 L j†L j, ρ(t) (Page 20).

Appendix C Quantum Information Measures: To quantify quantum coherence within the tryptophan (Trp) network and between its substructures, we employ the l1-norm of coherence as introduced in the resource theory of quantum coherence (Page 21). The logarithmic negativity, EN, is related to the negativity and serves as a good indicator of entanglement (Page 21).

Appendix D Supplementary Results: In Fig. 13 we summarize, for each embedding and initial state, the four site pairs with the largest time-averaged pair mutual information (Page 22). We evaluate how quantum coherence is shared between the excited tubulin (no. 2) and the remaining tubulins in a single spiral (Page 23). The trace distance Dk(t) = 1/2 Tr [ρ(1) X k(t) - ρ(2) X k(t)] quantifies memory through backflow, and the associated scalar non-Markovianity for X k is the total positive variation N (k) = Σ D˙ k(t)>0 D˙ k(t) dt (Page 23).

Appendix E Coherence Transfer from One Tubulin to a Spiral: Embedding reshapes the balance between internal circulation of correlations and outward loss, introducing oscillatory exchange with the surrounding tubulins (Page 12). Appendix F Time-Resolved Non-Markovian Backflow on Two Tubulin Subsystems (Single Spiral): Memory effects are relevant for neuronal microtubules because they show that correlations are not irreversibly lost to surrounding tubulins; the dimer or spiral can transiently store and return population and coherence to a local subsystem (Page 11). Appendix G Transition to Larger Assemblies: Ordered structures sustain long-range correlated coherence, while static or structural disorder localize the superradiant component, suppressing long-range quantum coherence (Page 13).

1. Active Information Routing based on Initial State Preparation: The improved AI system can predict how preparation shapes correlation generation by simulating five distinct initial states—superradiant eigenstates, subradiant eigenstates, maximally coherent uniform superposition, fully mixed uniform distribution, and site-localized injections—to determine the resulting direction and persistence of information flow.

2. Structural Condition Sensitivity Analysis: The system can assess the impact of physical environment on quantum information by quantifying how static energetic and structural disorder suppresses long-range transport in larger ordered lattices versus disordered ones, using metrics like the reduction in correlated coherence.

3. Memory Effect Detection via Non-Markovian Backflow: The AI system can detect transient information storage capabilities by monitoring memory effects through the quantification of trace distance backflow, specifically identifying when this occurs between subsystems like a focal tubulin and its neighbors in spiral structures.

4. Resource Quantification for Biological Relevance: The improved system can provide a quantitative language to assess functional relevance by distinguishing between ps–ns scale coherent redistribution and longer-scale radiative lifetimes, allowing researchers to identify which timescales overlap with downstream molecular degrees of freedom evolving on ns–ms scales.