PIATRA . INSTITUTE

Pettini tensor networks

tensor network compression of biological search processes with optional long-range electrodynamic recruitment

based on 2022, Pettini et al., Experimental evidence for long-distance electrodynamic intermolecular forces

ClaudeMarch 2026·initial implementation

diffusion-dominated regime

The system behaves mostly like conventional diffusion/sliding with weak long-range recruitment.

mobility: 61%

search: 89%

MPS chain · bond χ ~ 1.1 · 80 sites

probability distribution over DNA sites · protein starts at site 20 · target at site 60

sweep

coupling sweep · search time and compressibility vs coupling

resonance gain

0.1%

baseline mobility

61.0%

target bias

15.1%

compressibility

92.5%

search time

89.3%

sensitivity analysis · search time

3D diffusion

Δ0.302

1D sliding

Δ0.247

ionic noise

Δ0.180

resonance match

Δ0.004

activation

Δ0.004

coupling

Δ0.004

baseline: 0.893 · each parameter swept min→max while others held constant

Qualitative toy model — calibration error reflects intentional simplifications (no crowding, no DNA packaging, no conformational states).

t = 100%

From global to local

A generic quantum state of a chain of $N$ sites has $d^N$ coefficients — exponentially large data. Frank Verstraete’s key insight is that physically relevant states can be encoded by multiplying small local matrices:

c_{i_1 \ldots i_N} = \mathrm{Tr}\!\left(A^{i_1} A^{i_2} \cdots A^{i_N}\right)

Each matrix $A^{i_k}$ is chosen by the local state at site $k$ . The hidden matrix dimension (bond dimension $\chi$ ) stores the nontrivial correlations. This is a matrix product state (MPS) — the simplest tensor network.

Biology version

The same compression logic applies to biological search. A protein searching for a target on DNA has a joint state over position, diffusion mode, conformation, and the local environment at every DNA site. The full probability distribution grows exponentially with the number of sites. A tensor-train approximation replaces this with local tensors:

P_t(x,m,c,\varphi;\, d_1,\ldots,d_N) \approx \sum_{\alpha} A^{[\text{p}]}_\alpha(x,m,c,\varphi)\, A^{[1]}_\alpha(d_1) \cdots A^{[N]}_\alpha(d_N)

The protein state includes position $x$ , diffusion mode $m$ (3D diffusion, 1D sliding, or bound), conformation $c$ , and vibrational activation $\varphi$ . Each DNA site $n$ carries a motif class $\sigma_n$ , frequency bin $\omega_n$ , and hydration state $h_n$ .

The search generator

The search process evolves under a stochastic generator with four terms:

\partial_t P_t = \mathcal{L}\, P_t, \qquad \mathcal{L} = \mathcal{L}_{\text{diff}} + \mathcal{L}_{\text{slide}} + \mathcal{L}_{\text{chem}} + \mathcal{L}_{\text{ED}}

The first three terms are standard: free 3D diffusion in solvent, 1D sliding along nearby DNA, and short-range chemical recognition. The fourth term $\mathcal{L}_{\text{ED}}$ is a Pettini-inspired long-range electrodynamic recruitment that biases the protein toward frequency-compatible target sites.

Pettini interaction term

The electrodynamic coupling is modeled as a distance-dependent attraction that activates only when there is vibrational activation, coupling strength, and frequency compatibility:

V_{\text{ED}}(x,n) = -\, g \cdot \varphi \cdot \chi_p(c) \cdot \chi_n(\omega_n, h_n) \cdot S(\Delta\omega) \;/\; (r^3 + \varepsilon)

Here $g$ is the coupling strength, $\varphi$ the activation state, $\chi_p$ and $\chi_n$ are susceptibility functions, $S(\Delta\omega)$ is the spectral overlap, and the $r^{-3}$ dependence reflects dipolar interaction decay. The $\varepsilon$ regularizer prevents divergence at zero distance.

What is solid, what is speculative

Tensor-network compression of high-dimensional biological state spaces is mathematically well-founded. Reaction networks, master equations, and structured probabilistic systems are natural targets for these methods. The computational framework is credible today.

The Pettini-style electrodynamic mechanism — sustained low-frequency collective modes biasing molecular encounters over long distances — remains an open research question. The evidence is stronger in controlled in vitro settings than in living cells, where crowding, ionic screening, and thermal noise present significant challenges.

Model changelog

v1March 2026

Initial 6-parameter toy model with diffusion, sliding, and resonance terms
Probability distribution visualization over 80 DNA sites
Coupling sweep showing search time / compressibility tradeoff
Qualitative metrics: resonance gain, baseline mobility, target bias, compressibility, search time

read the research companion →