Constant Frame-Tension Box Rescaling Algorithm

PositionRescaleFrameTensionCoupling samples a membrane under a prescribed frame tension by allowing the dimensions of the periodic simulation box to fluctuate.

For a quasi-planar membrane with periodic boundaries in the $x$ and $y$ directions, the projected membrane area is

A_p = L_x L_y .

At fixed projected area, the box dimensions are constant. In the constant-frame-tension ensemble, $A_p$ is allowed to fluctuate and is coupled to an externally imposed frame tension $\tau$.

The method follows the constant-frame-tension Monte Carlo scheme described by Pezeshkian and Ipsen, where a trial box rescaling is accepted with an additional mechanical-work contribution and a Jacobian arising from the rescaling of all vertex coordinates.

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Trial Box Move

For an $xy$ frame, a trial deformation is generated as

L_x' = \lambda_x L_x,
\qquad
L_y' = \lambda_y L_y,

while $L_z$ remains unchanged.

In the isotropic implementation, the aspect ratio is preserved. A small change $\Delta L_x$ is selected and

\Delta L_y
=
\Delta L_x \frac{L_y}{L_x},

so that

\frac{L_x'}{L_y'}
=
\frac{L_x}{L_y}.

The corresponding scaling factors are

\lambda_x
=
1+\frac{\Delta L_x}{L_x},
\qquad
\lambda_y
=
1+\frac{\Delta L_y}{L_y}.

All vertex coordinates are rescaled together with the box:

x_i' = \lambda_x x_i,
\qquad
y_i' = \lambda_y y_i.

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Thermodynamic Ensemble

Let a membrane configuration be denoted by

X = \{\mathbf r_i\},

with internal energy

E(X;A_p).

At fixed projected area, the configurational partition function is

Z(A_p)
=
\int_{A_p} dX\,
e^{-\beta E(X)},

where

\beta = \frac{1}{k_B T}.

To sample a membrane at prescribed frame tension $\tau$, the projected area becomes a fluctuating thermodynamic variable.

The corresponding effective thermodynamic potential is

\mathcal{H}_\tau
=
E-\tau A_p

and therefore

Z_\tau
=
\int dA_p
\int_{A_p} dX\,
\exp\left[
-\beta(E-\tau A_p)
\right]

For a trial box move,

A_p \rightarrow A_p',

the change in the effective energy is

\Delta \mathcal{H}_\tau
=
\Delta E
-
\tau \Delta A_p

with

\Delta E = E'-E

and

\Delta A_p = A_p'-A_p.

This is implemented in FreeDTS as

tot_diff_energy -= m_SigmaP *
                   (new_systemsize - old_systemsize);

For an XY move,

old_systemsize = Lx * Ly;
new_systemsize = Lx_new * Ly_new;

and therefore

\Delta E_{\rm ext}
=
-\tau(A_p'-A_p).

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Why There Is a Jacobian

The mechanical-work contribution alone is not sufficient because changing the box also changes the integration measure of the vertex coordinates.

For an $xy$ rescaling,

x_i' = \lambda_x x_i,
\qquad
y_i' = \lambda_y y_i.

For one vertex,

dx_i' dy_i'
=
\lambda_x \lambda_y\,
dx_i dy_i.

For $N$ vertices,

dX'
=
(\lambda_x\lambda_y)^N dX.

Since

\lambda_x\lambda_y
=
\frac{L_x'L_y'}{L_xL_y}
=
\frac{A_p'}{A_p},

the Jacobian is

J
=
\left(
\frac{A_p'}{A_p}
\right)^N
.

Detailed balance therefore gives

\frac{P(X\rightarrow X')}
     {P(X'\rightarrow X)}
=
\left(
\frac{A_p'}{A_p}
\right)^N
\exp
\left[
-\beta
\left(
\Delta E-\tau\Delta A_p
\right)
\right]

The Metropolis acceptance probability becomes

P_{\rm acc}
=
\min
\left[
1,
\left(
\frac{A_p'}{A_p}
\right)^N
\exp
\left(
-\beta\Delta E
+
\beta\tau\Delta A_p
\right)
\right]

This is the constant-frame-tension acceptance probability used in the original formulation.

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Logarithmic Form Used in FreeDTS

FreeDTS evaluates the acceptance probability in logarithmic form:

const double logJacobian =
    NV * (std::log(lx)
        + std::log(ly)
        + std::log(lz));

const double logAcceptance =
    logJacobian
    - m_Beta * tot_diff_energy
    + m_DBeta;

if (std::log(temp) < logAcceptance) {
    // accept move
}

For an XY move,

\lambda_z = 1,

and therefore

\log J
=
N
\left[
\log\lambda_x
+
\log\lambda_y
\right]

Using

\lambda_x\lambda_y
=
\frac{A_p'}{A_p},

we obtain

\log J
=
N
\log
\left(
\frac{A_p'}{A_p}
\right)

Hence

\log P
=
N
\log
\left(
\frac{A_p'}{A_p}
\right)
-
\beta
\left(
\Delta E-\tau\Delta A_p
\right)

This is exactly the logarithmic form of the constant-frame-tension Metropolis criterion.

Reference

W. Pezeshkian and J. H. Ipsen, Fluctuations and conformational stability of a membrane patch with curvature inducing inclusions, Soft Matter 15, 9974–9981 (2019).