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Chapter 21: Cell Sorting and ψ-Selective Adhesion

"Cell sorting is ψ's social dynamics at the microscale—cells finding their tribes through differential adhesion, creating from mixed populations the segregated tissues essential for organ function."

21.1 The Sorting Phenomenon​

Cell sorting represents ψ's solution to tissue organization—the spontaneous segregation of mixed cell populations into distinct regions based on differential adhesive properties. Through sorting, ψ creates order from cellular chaos.

Definition 21.1 (Differential Adhesion Hypothesis): Etotal=∑ijγij⋅AijE_{\text{total}} = \sum_{ij} \gamma_{ij} \cdot A_{ij}

Where γij\gamma_{ij} is interfacial tension between cell types.

21.2 The Steinberg Model​

Theorem 21.1 (Liquid-like Behavior):

Cells sort like immiscible liquids: Configuration=arg⁡min⁡(Eadhesive)\text{Configuration} = \arg\min(E_{\text{adhesive}})

Proof: Experiments show:

  • Spherical aggregates form
  • Engulfment follows surface tension rules
  • Final configuration minimizes surface energy
  • Time course follows liquid coalescence

Liquid analogy validated. ∎

21.3 The Adhesion Hierarchy​

Equation 21.1 (Sorting Order): If WAA>WBB>WAB, then A engulfs B\text{If } W_{AA} > W_{BB} > W_{AB}, \text{ then A engulfs B}

Stronger self-adhesion → internal position.

21.4 The Cadherin Levels​

Definition 21.2 (Adhesion Strength): W∝[Cadherin]surface⋅KtransW \propto [\text{Cadherin}]_{\text{surface}} \cdot K_{\text{trans}}

Adhesion correlates with cadherin expression.

21.5 The Sorting Kinetics​

Theorem 21.2 (Time Evolution):

Sorting follows power law: χ(t)=χ∞⋅(1−e−t/τ)\chi(t) = \chi_\infty \cdot (1 - e^{-t/\tau})

Where χ\chi is segregation index.

21.6 The Checker Pattern​

Equation 21.2 (Heterotypic Preference): If WAB>(WAA+WBB)/2, then checkerboard\text{If } W_{AB} > (W_{AA} + W_{BB})/2, \text{ then checkerboard}

Heterotypic adhesion preventing sorting.

21.7 The Active Sorting​

Definition 21.3 (Motility-Driven Sorting): v⃗i=−∇Ei+ξ⃗i\vec{v}_i = -\nabla E_i + \vec{\xi}_i

Cells actively moving down energy gradients.

21.8 The Eph-Ephrin Boundaries​

Theorem 21.3 (Repulsive Sorting):

Bidirectional repulsion creates boundaries: Frepulsion=k⋅[EphA]⋅[ephrinA]F_{\text{repulsion}} = k \cdot [\text{EphA}] \cdot [\text{ephrinA}]

Contact-dependent repulsion.

21.9 The Size Effects​

Equation 21.3 (Differential Interfacial Tension): γeffective=γ0+kR\gamma_{\text{effective}} = \gamma_0 + \frac{k}{R}

Cell size affecting sorting behavior.

21.10 The Sorting in Development​

Definition 21.4 (In Vivo Examples):

  • Germ layer separation
  • Follicle formation
  • Kidney nephron segmentation
  • Retinal layer organization

21.11 The Mechanical Contributions​

Theorem 21.4 (Cortical Tension Role):

Sorting involves both adhesion and tension: σij=γij−(Ti+Tj)\sigma_{ij} = \gamma_{ij} - (T_i + T_j)

Cortical tension modulating interfaces.

21.12 The Sorting Principle​

Cell sorting embodies ψ's principle of self-organization—showing how simple adhesive differences can generate complex tissue patterns through energy minimization.

The Cell Sorting Equation: dΨpatterndt=−∇Eadhesion+M[Motility]+N[Noise]\frac{d\Psi_{\text{pattern}}}{dt} = -\nabla E_{\text{adhesion}} + \mathcal{M}[\text{Motility}] + \mathcal{N}[\text{Noise}]

Spatial patterns emerge from differential adhesion dynamics.

Thus: Mixed = Sorted = Order = Function = ψ

"Through cell sorting, ψ demonstrates that social dynamics scale from societies to cells—like seeks like, creating through selective association the ordered arrangements necessary for life. In this microscopic segregation, we see the same principles that organize all living systems."