Theory Background
=================

This section provides the theoretical foundation for the microscopic gating model,
derived from first principles in statistical mechanics.

Grand Canonical Framework
-------------------------

The model treats the bulk TC (third component) concentration :math:`\phi` as a
chemical potential reservoir (ideal reservoir approximation):

.. math::

    z \equiv e^{\beta\mu} \simeq \phi/\phi^\circ, \qquad \beta = (k_B T)^{-1}

where :math:`z` is the fugacity, :math:`\mu` is the chemical potential, and
:math:`\phi^\circ` is a reference concentration for nondimensionalization.

Langmuir Adsorption Isotherm
----------------------------

For a single binding site, the grand canonical partition function with two states
(empty/occupied) yields:

.. math::

    Z_i = 1 + z e^{\beta\epsilon}

where :math:`\epsilon > 0` is the binding energy. The occupancy fraction is:

.. math::

    \theta(\phi) = \frac{z e^{\beta\epsilon}}{1 + z e^{\beta\epsilon}}
                 = \frac{\phi/K_d}{1 + \phi/K_d}

with dissociation constant :math:`K_d \equiv \phi^\circ e^{-\beta\epsilon}`.

For cooperative binding, the Hill isotherm generalizes this to:

.. math::

    \theta(\phi) = \frac{(\phi/K_d)^n}{1 + (\phi/K_d)^n}

where :math:`n` is the Hill coefficient (:math:`n=1` recovers Langmuir).

Symmetric Gating Function
-------------------------

The key insight of the model is that productive bridge formation requires:

1. Particle site occupied by TC: probability :math:`\theta_P`
2. Network site **unoccupied** (to avoid blocking): probability :math:`1-\theta_N`
3. Geometric contact within capture volume: factor :math:`\langle\chi\rangle`
4. Successful closure given contact: probability :math:`\kappa_B`

For symmetric binding (:math:`K_P \approx K_N \approx K_d`, hence
:math:`\theta_P \approx \theta_N \approx \theta`), the bridge probability per
site pair is:

.. math::

    p_B(\phi) = \langle\chi\rangle \kappa_B \, \theta(\phi)[1-\theta(\phi)]

The gating function :math:`\mathcal{G}(\phi) = \theta(1-\theta)` has a
characteristic bell shape:

.. math::

    \mathcal{G}(\phi) = \frac{\phi/K_d}{(1+\phi/K_d)^2}

with peak at :math:`\phi^\ast = K_d` and maximum value :math:`\mathcal{G}_{\max} = 1/4`.

**Physical interpretation:**

- **Low concentration** (:math:`\phi \to 0`): Bridge formation limited by
  "starvation" (:math:`\theta \approx \phi/K_d \to 0`)
- **High concentration** (:math:`\phi \to \infty`): Bridge formation limited by
  "blocking" (:math:`1-\theta \to 0` as sites become saturated)
- **Intermediate concentration**: Optimal balance yields maximum bridging

Multivalency Statistics
-----------------------

For a particle with :math:`M` binding sites and :math:`N_{\text{acc}}` accessible
network sites within a pore, the total number of potential bridge site pairs is:

.. math::

    N_{\text{pair}} = M \cdot N_{\text{acc}}

Under the independent-pair approximation (valid when :math:`p_B` is small or
site-site coupling is weak), the bridge count :math:`n_b` follows a binomial
distribution:

.. math::

    P(n_b) = \binom{N_{\text{pair}}}{n_b} p_B^{n_b} (1-p_B)^{N_{\text{pair}}-n_b}

In the Poisson limit (:math:`N_{\text{pair}} \to \infty`, :math:`p_B \to 0`,
with :math:`\lambda = N_{\text{pair}} p_B` finite):

.. math::

    P(n_b) \approx \frac{\lambda^{n_b} e^{-\lambda}}{n_b!}

where the Poisson intensity is:

.. math::

    \lambda(\phi) = N_{\text{pair}} \langle\chi\rangle \kappa_B \, \mathcal{G}(\phi)
                  \equiv \lambda_0 \, \mathcal{G}(\phi)

Gate Open/Closed Probabilities
------------------------------

The gate state is defined by bridge presence:

- **Gate Open**: No bridges (:math:`n_b = 0`)
- **Gate Closed**: At least one bridge (:math:`n_b \geq 1`)

Under Poisson statistics:

.. math::

    P_{\text{open}}(\phi) = e^{-\lambda(\phi)}

    P_{\text{closed}}(\phi) = 1 - e^{-\lambda(\phi)}

Free Energy Landscape
---------------------

The effective 1D free energy landscape for a particle in a cage with :math:`n_b`
bridges is:

.. math::

    F(x; n_b) = \frac{1}{2}\kappa x^2 + n_b \, u_b(x)

where:

- :math:`x`: Reaction coordinate (radial displacement from cage center)
- :math:`\kappa`: Cage stiffness (elastic confinement)
- :math:`u_b(x)`: Single-bond potential (saturating/breakable, e.g., Morse)

For a Morse bond potential:

.. math::

    u_b(x) = \epsilon_b \left(1 - e^{-x/\ell_b}\right)^2

with bond depth :math:`\epsilon_b` and characteristic length :math:`\ell_b`.

Kramers Escape Rate
-------------------

In the overdamped limit, the escape rate for fixed :math:`n_b` is given by
Kramers-Smoluchowski theory:

.. math::

    k_{\text{esc}}(n_b) = \frac{\sqrt{F''(x_0;n_b)|F''(x^\ddagger;n_b)|}}{2\pi\gamma}
                          \exp\left[-\beta \Delta F^\ddagger(n_b)\right]

where:

- :math:`x_0`: Local minimum position (cage center)
- :math:`x^\ddagger`: Saddle point position (barrier top)
- :math:`\gamma`: Effective friction
- :math:`\Delta F^\ddagger(n_b) = F(x^\ddagger;n_b) - F(x_0;n_b)`: Barrier height

Using the pore-neck approximation (barrier defined by geometric escape coordinate
:math:`x_c`):

.. math::

    \Delta F^\ddagger(n_b) \approx \frac{1}{2}\kappa x_c^2 + n_b \epsilon_{\text{eff}}

where :math:`\epsilon_{\text{eff}} = u_b(x_c)` is the effective bond penalty at
the escape coordinate.

Poisson-Averaged Escape Rate
----------------------------

The physically observable escape rate is averaged over the bridge count
distribution:

.. math::

    k_{\text{esc}}(\phi) = \sum_{n_b=0}^\infty P(n_b|\phi) \, k_{\text{esc}}(n_b)

For the exponential ansatz :math:`k_{\text{esc}}(n_b) = \tilde{k}_0 e^{-\alpha n_b}`
with :math:`\alpha = \beta\epsilon_{\text{eff}}`, this yields the closed form:

.. math::

    k_{\text{esc}}(\phi) = \tilde{k}_0 \exp\left[-\lambda(\phi)(1 - e^{-\alpha})\right]

Effective Diffusion Coefficient
-------------------------------

Via jump diffusion mapping (step length :math:`\ell`, dimension :math:`d=3`):

.. math::

    D_{\text{eff}}(\phi) = \frac{\ell^2}{6} k_{\text{esc}}(\phi)
                         = D_{\text{free}} \exp\left[-\lambda(\phi)(1 - e^{-\beta\epsilon_{\text{eff}}})\right]

This exhibits **re-entrant non-monotonicity** due to the bell-shaped
:math:`\lambda(\phi)`:

- Diffusion is **suppressed** at intermediate concentrations where bridging is maximal
- Diffusion recovers at both low and high concentration limits

Entropic Widening (Network Softening)
-------------------------------------

TC binding can induce network plasticization, leading to concentration-dependent
renormalization of cage parameters:

.. math::

    \kappa(\phi) = \kappa_0 \left[1 - \chi_\kappa \mathcal{G}(\phi)\right]

where :math:`\chi_\kappa \geq 0` is the softening strength (stability requires
:math:`\chi_\kappa < 4` for Langmuir gating).

This produces a competing **enhancement** mechanism:

.. math::

    D_{\text{eff}}(\phi) = D_\star \exp\left[(A - \Gamma)\mathcal{G}(\phi)\right]

with:

.. math::

    A = \beta \frac{1}{2} \kappa_0 x_c^2 \chi_\kappa \quad \text{(widening gain)}

    \Gamma = \lambda_0 (1 - e^{-\beta\epsilon_{\text{eff}}}) \quad \text{(trapping loss)}

**Transport regime criterion:**

- :math:`A > \Gamma`: **Enhancement-dominated** (bell-shaped, faster at intermediate :math:`\phi`)
- :math:`A < \Gamma`: **Suppression-dominated** (U-shaped, slower at intermediate :math:`\phi`)
- :math:`A = \Gamma`: **Critical line** (nearly concentration-independent)

Capture Island Phase Boundaries
-------------------------------

The "gated regime" or "capture island" is defined by timescale bracketing:

.. math::

    \tau_{\text{net}} \lesssim \tau_{\text{esc}}(\phi) \lesssim \tau_{\text{obs}}

For the suppression regime (:math:`\Gamma > A`), this yields analytic concentration
boundaries. Given threshold :math:`g \in (0, 1/4]`, solving
:math:`\mathcal{G}(\phi) = g` for Langmuir gives:

.. math::

    \phi_{\pm}(g) = K_d \cdot \frac{1 - 2g \pm \sqrt{1 - 4g}}{2g}

The capture island corresponds to :math:`\phi \in [\phi_-(g_+), \phi_+(g_+)]` where
:math:`g_\pm` are determined by the time thresholds.

Parameter Dictionary
--------------------

The following table maps model parameters to their physical meanings and
experimental determination methods:

.. csv-table:: Model Parameters
   :header: "Parameter", "Physical Meaning", "Measurement/Estimation"
   :widths: 15, 35, 35

   ":math:`K_d`", "Dissociation constant", "Adsorption isotherm (SPR/ITC)"
   ":math:`\theta`", "Site occupancy fraction", "From :math:`K_d` and :math:`\phi`"
   ":math:`\langle\chi\rangle`", "Geometric contact probability", "Pore geometry, simulation"
   ":math:`\kappa_B`", "Closure success probability", "Fit from peak amplitude"
   ":math:`M`", "Particle valency (binding sites)", "Known from particle design"
   ":math:`N_{\text{acc}}`", "Accessible network sites", "Pore size, chain density estimate"
   ":math:`\lambda_0`", "Bridge statistics strength", "Fit from :math:`D_{\text{eff}}`"
   ":math:`\epsilon_{\text{eff}}`", "Effective bond free energy", "Fit from concentration dependence"
   ":math:`\kappa_0`", "Baseline cage stiffness", "Short-time MSD plateau"
   ":math:`\chi_\kappa`", "Softening strength", "Fit from :math:`\kappa(\phi)`"
   ":math:`x_c`", "Escape coordinate (pore neck)", "Pore geometry estimate"
   ":math:`\ell`", "Jump length (pore-to-pore)", "Network structure estimate"

Key References
--------------

- Eq. (S3)-(S4): Langmuir isotherm from grand canonical ensemble
- Eq. (S5)-(S8): Symmetric gating function derivation
- Eq. (S9)-(S14): Bridge count statistics and Poisson limit
- Eq. (S18)-(S30): Free energy landscape and barrier height
- Eq. (S32)-(S37): Kramers escape rate with prefactor
- Eq. (S41)-(S45): Poisson-averaged escape rate and diffusion
- Eq. (S64)-(S74): Entropic widening and competition criterion
- Eq. (S77)-(S83): Unified phase diagram