Phase Boundaries

This module implements models for analyzing phase boundaries and capture islands in the parameter space of the gating system.

Dome Phase Boundaries

class microscopic_gating.phase_boundaries.DomePhaseBoundaries(K, n=1.0)

Bases: object

Analytic concentration boundaries for the bell-shaped gating function.

For Langmuir (and Hill via substitution t=(phi/K)^n), the symmetric gating is:

\[\mathcal G(\phi)=\theta(1-\theta)=\frac{t}{(1+t)^2},\quad t=(\phi/K)^n\]

Given a threshold \(g\in(0,1/4]\), solving \(\mathcal G=g\) yields:

\[t_{\pm}(g)=\frac{1-2g\pm\sqrt{1-4g}}{2g}\]

hence:

\[\phi_{\pm}(g)=K\,t_{\pm}(g)^{1/n}. \quad (S66\ \text{and Hill extension})\]

Parameters:

• K (float) – Dissociation constant K_d (same unit as phi).

• n (float) – Hill coefficient. n=1 reduces to Langmuir.

Notes

• Solutions exist only for g <= 1/4 (peak value).

• For n=1, symmetry implies \(\phi_-(g)\phi_+(g)=K^2\).

K: float

n: float = 1.0

t_solutions(g)

Solve for t in g = t/(1+t)^2.

Parameters:

g (float) – Threshold in (0, 1/4].

Returns:

Two positive solutions with t_minus <= 1 <= t_plus.

Return type:

t_minus, t_plus

phi_solutions(g)

Solve for phi in G(phi)=g.

Parameters:

g (float) – Threshold in (0, 1/4].

Returns:

Two solutions with phi_minus <= K <= phi_plus.

Return type:

phi_minus, phi_plus

__init__(K, n=1.0)

Parameters:

• K (float) –

• n (float) –

Return type:

None

Capture Island Window

class microscopic_gating.phase_boundaries.CaptureIslandWindow(lambda0, boundaries)

Bases: object

Compute capture-island (dome-like) concentration intervals from time-scale windows.

The strict gated window is:

\[\tau_{\text{net}} \lesssim \tau(\phi) \lesssim \tau_{\text{obs}}\]

Using either:

• Rate-averaged \(\tau_{\text{eff}}\) (S53) => lambda bounds (S60)

• Time-averaged \(\langle\tau\rangle\) (S69) => lambda bounds (S70)

With:

\[\lambda(\phi) = \lambda_0 \mathcal{G}(\phi)\]

Therefore:

\[\begin{split}\mathcal{G}(\phi) \in [g_-, g_+], \quad g_{\\pm} = \lambda_{\\pm} / \lambda_0\end{split}\]

Because \(G(\phi)\) is bell-shaped, the set \(\\{\phi: G \in [g_-, g_+]\\}\) is typically two disjoint intervals flanking the peak (an annulus/band around \(\phi \approx K\)).

Parameters:

• lambda0 (float) – Geometric/multivalency strength parameter \(\lambda_0\).

• boundaries (DomePhaseBoundaries) – Analytic dome boundary solver for \(G(\phi) = g\).

See also

DomePhaseBoundaries

Solver for bell-shaped gating function boundaries.

lambda0: float

boundaries: DomePhaseBoundaries

exists(g_minus)

Check whether an island exists (i.e., threshold below peak).

Parameters:

g_minus (float) – Lower gating threshold \(g_-\).

Returns:

exists – True if \(g_- \leq 1/4\) (the maximum of the gating function).

Return type:

bool

g_from_lambda(lam)

Convert lambda threshold to g threshold.

Parameters:

lam (float) – Lambda threshold \(\lambda\).

Returns:

g – Gating threshold \(g = \lambda / \lambda_0\).

Return type:

float

Raises:

ValueError – If lambda0 is not positive.

phi_intervals_for_g_band(g_minus, g_plus)

Return concentration intervals where \(G(\phi)\) is between \([g_-, g_+]\).

Parameters:

• g_minus (float) – Lower threshold (>=0).

• g_plus (float) – Upper threshold (>= g_minus).

Returns:

intervals – List of (phi_left, phi_right). Usually two disjoint intervals.

Return type:

list of tuple

Notes

For \(0 < g_- < g_+ \leq 1/4\):

• Outer interval from \(g_-\): \([\phi_-(g_-), \phi_+(g_-)]\)

• Inner interval from \(g_+\): \([\phi_-(g_+), \phi_+(g_+)]\)

The band is: \([\phi_-(g_-), \phi_-(g_+)] \cup [\phi_+(g_+), \phi_+(g_-)]\).

__init__(lambda0, boundaries)

Parameters:

• lambda0 (float) –

• boundaries (DomePhaseBoundaries) –

Return type:

None

Theory Background

Dome-Shaped Phase Boundaries

The gating function \(G(\phi) = \theta(1-\theta)\) produces a dome-shaped dependence on concentration. Phase boundaries occur where:

\[G(\phi) = G_{\text{threshold}}\]

This yields two solutions (low and high concentration boundaries) that define the “dome” region where gating is effective.

Capture Islands

Capture islands are regions in parameter space where the system exhibits specific transport behavior. The window of concentrations supporting capture depends on:

• Binding affinity \(K\)

• Site pair counts \(N_{\text{pair}}\)

• Geometric factors \(\chi\)

• Chemical factors \(\kappa_B\)