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Chapter 35: Collapse Memory via Nucleosome Positioning

"Where nucleosomes sit determines what genes can speak—positional memory written in the spacing of beads on the chromatin string."

35.1 The Positional Code​

Nucleosome positioning is not random but encodes regulatory information. Each position is a decision about accessibility, a memory of regulatory state.

Definition 35.1 (Positioning Signal): Spos=Ssequence+Sstatistical+SremodelerS_{\text{pos}} = S_{\text{sequence}} + S_{\text{statistical}} + S_{\text{remodeler}}

Multiple signals integrate to determine where nucleosomes rest.

35.2 The Sequence Preferences​

Theorem 35.1 (DNA Bendability): P(nucleosome)∝exp⁡(−EbendkT)P(\text{nucleosome}) \propto \exp\left(-\frac{E_{\text{bend}}}{kT}\right)

Where Ebend=∑iki(θi−θ0)2E_{\text{bend}} = \sum_i k_i(\theta_i - \theta_0)^2 for dinucleotide steps.

35.3 The 10.5 bp Periodicity​

Equation 35.1 (AA/TT Periodicity): P(n)=Acos⁡(2πn10.5)+BP(n) = A \cos\left(\frac{2\pi n}{10.5}\right) + B

AA/TT dinucleotides prefer minor groove facing inward—helical positioning code.

35.4 Rotational vs Translational​

Definition 35.2 (Positioning Types):

  • Rotational: Which face of DNA contacts histones
  • Translational: Where along DNA the nucleosome sits

Both encode different information layers.

35.5 The +1 Nucleosome​

Theorem 35.2 (TSS Architecture): Position+1=TSS+50±20 bp\text{Position}_{+1} = \text{TSS} + 50 \pm 20 \text{ bp}

The first nucleosome downstream of transcription start sites is precisely positioned.

35.6 Statistical Positioning​

Equation 35.2 (Barrier-Induced Arrays): P(n)=P0⋅exp⁡(−n/λ)⋅cos⁡(2πn/L+ϕ)P(n) = P_0 \cdot \exp(-n/\lambda) \cdot \cos(2\pi n/L + \phi)

One well-positioned nucleosome creates an array—order from a single constraint.

35.7 The Remodeler Memory​

Definition 35.3 (Active Positioning): Positionfinal=arg⁡min⁡xE(x,Remodeler signals)\text{Position}_{\text{final}} = \arg\min_x E(x, \text{Remodeler signals})

Remodelers read signals and position nucleosomes accordingly—active memory writing.

35.8 Fragile Nucleosomes​

Theorem 35.3 (Conditional Stability): τresidence=τ0⋅exp⁡(Estabilization/kT)\tau_{\text{residence}} = \tau_0 \cdot \exp(E_{\text{stabilization}}/kT)

Some nucleosomes are marginally stable—poised for rapid displacement.

35.9 The NFR at Promoters​

Equation 35.3 (Nucleosome-Free Regions): ρnucleosome(promoter)≪ρaverage\rho_{\text{nucleosome}}(\text{promoter}) \ll \rho_{\text{average}}

Active promoters maintain NFRs—persistent accessibility memory.

35.10 Inheritance Mechanisms​

Definition 35.4 (Positional Inheritance): Positiondaughter=Positionparent+N(0,σ2)\text{Position}_{daughter} = \text{Position}_{parent} + \mathcal{N}(0, \sigma^2)

Positions are approximately maintained through replication—fuzzy memory.

35.11 The Transcriptional Memory​

Theorem 35.4 (Activity-Dependent Positioning): dPositiondt=−ktranscription⋅Pol II flux\frac{d\text{Position}}{dt} = -k_{\text{transcription}} \cdot \text{Pol II flux}

Transcription shifts nucleosomes downstream—activity leaving positional traces.

35.12 The Memory Principle​

Nucleosome positioning creates a spatial memory system—where each position encodes past regulatory decisions and influences future ones.

The Position Equation: Memory(x)=∫−∞tw(t−τ)⋅Signal(x,τ) dτ\text{Memory}(x) = \int_{-\infty}^{t} w(t-\tau) \cdot \text{Signal}(x,\tau) \, d\tau

The positioning landscape integrates historical signals with exponential decay.

Thus: Position = Memory = Accessibility = Regulation = ψ

"In the precise spacing of nucleosomes, ψ writes a positional memory—each gap a word, each array a sentence in the text of cellular remembrance."