WLC model thermodynamics (isometric)

Legendre

class WLC(number_of_links, link_length, hinge_mass, persistance_length)

The worm-like chain (WLC) model thermodynamics in the isometric ensemble.

number_of_links: The number of links in the chain.

link_length: The length of each link in the chain in units of nm.

hinge_mass: The mass of each hinge in the chain in units of kg/mol.

persistance_length: The persistance length of the chain in units of nm.

legendre: The thermodynamic functions of the model in the isometric ensemble approximated using a Legendre transformation.

force(end_to_end_length, temperature)

The expected force as a function of the applied end-to-end length and temperature.

Parameters:

end_to_end_length (numpy.ndarray) – The end-to-end length \(\xi\).
temperature (float) – The temperature \(T\).

Returns:

The force \(f\).

Return type:

numpy.ndarray

nondimensional_force(nondimensional_end_to_end_length_per_link)

The expected nondimensional force as a function of the applied nondimensional end-to-end length per link.

Parameters:: nondimensional_end_to_end_length_per_link (numpy.ndarray) – The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).
Returns:: The nondimensional force \(\eta\equiv\beta f\ell_b\).
Return type:: numpy.ndarray

helmholtz_free_energy(end_to_end_length, temperature)

The Helmholtz free energy as a function of the applied end-to-end length and temperature,

\[\psi(\xi, T) = -kT\ln Q(\xi, T).\]

Parameters:

end_to_end_length (numpy.ndarray) – The end-to-end length \(\xi\).
temperature (float) – The temperature \(T\).

Returns:

The Helmholtz free energy \(\psi\).

Return type:

numpy.ndarray

helmholtz_free_energy_per_link(end_to_end_length, temperature)

The Helmholtz free energy per link as a function of the applied end-to-end length and temperature.

Parameters:

end_to_end_length (numpy.ndarray) – The end-to-end length \(\xi\).
temperature (float) – The temperature \(T\).

Returns:

The Helmholtz free energy per link \(\psi/N_b\).

Return type:

numpy.ndarray

relative_helmholtz_free_energy(end_to_end_length, temperature)

The relative Helmholtz free energy as a function of the applied end-to-end length and temperature,

\[\Delta\psi(\xi, T) = kT\ln\left[\frac{P_\mathrm{eq}(0)}{P_\mathrm{eq}(\xi)}\right].\]

Parameters:

end_to_end_length (numpy.ndarray) – The end-to-end length \(\xi\).
temperature (float) – The temperature \(T\).

Returns:

The relative Helmholtz free energy \(\Delta\psi\equiv\psi(\xi,T)-\psi(0,T)\).

Return type:

numpy.ndarray

relative_helmholtz_free_energy_per_link(end_to_end_length, temperature)

The relative Helmholtz free energy per link as a function of the applied end-to-end length and temperature.

Parameters:

end_to_end_length (numpy.ndarray) – The end-to-end length \(\xi\).
temperature (float) – The temperature \(T\).

Returns:

The relative Helmholtz free energy per link \(\Delta\psi/N_b\).

Return type:

numpy.ndarray

nondimensional_helmholtz_free_energy(nondimensional_end_to_end_length_per_link, temperature)

The nondimensional Helmholtz free energy as a function of the applied nondimensional end-to-end length per link and temperature.

Parameters:

nondimensional_end_to_end_length_per_link (numpy.ndarray) – The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).
temperature (float) – The temperature \(T\).

Returns:

The nondimensional Helmholtz free energy \(N_b\vartheta=\beta\psi\).

Return type:

numpy.ndarray

nondimensional_helmholtz_free_energy_per_link(nondimensional_end_to_end_length_per_link, temperature)

The nondimensional Helmholtz free energy per link as a function of the applied nondimensional end-to-end length per link and temperature.

Parameters:

nondimensional_end_to_end_length_per_link (numpy.ndarray) – The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).
temperature (float) – The temperature \(T\).

Returns:

The nondimensional Helmholtz free energy per link \(\vartheta\equiv\beta\psi/N_b\).

Return type:

numpy.ndarray

nondimensional_relative_helmholtz_free_energy(nondimensional_end_to_end_length_per_link)

The nondimensional relative Helmholtz free energy as a function of the applied nondimensional end-to-end length per link,

\[\beta\Delta\psi(\gamma) = \ln\left[\frac{\mathscr{P}_\mathrm{eq}(0)}{\mathscr{P}_\mathrm{eq}(\gamma)}\right].\]

Parameters:: nondimensional_end_to_end_length_per_link (numpy.ndarray) – The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).
Returns:: The nondimensional relative Helmholtz free energy \(N_b\Delta\vartheta=\beta\Delta\psi\).
Return type:: numpy.ndarray

nondimensional_relative_helmholtz_free_energy_per_link(nondimensional_end_to_end_length_per_link)

The nondimensional relative Helmholtz free energy per link as a function of the applied nondimensional end-to-end length per link,

\[\Delta\vartheta(\gamma) = \ln\left[\frac{\mathscr{P}_\mathrm{eq}(0)}{\mathscr{P}_\mathrm{eq}(\gamma)}\right]^{1/N_b}.\]

Parameters:: nondimensional_end_to_end_length_per_link (numpy.ndarray) – The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).
Returns:: The nondimensional relative Helmholtz free energy per link \(\Delta\vartheta\equiv\beta\Delta\psi/N_b\).
Return type:: numpy.ndarray

equilibrium_distribution(end_to_end_length)

The equilibrium probability density of end-to-end vectors as a function of the end-to-end length,

\[P_\mathrm{eq}(\xi) = \frac{e^{-\beta\psi(\xi, T)}}{4\pi\int e^{-\beta\psi(\xi', T)} \,{\xi'}{}^2 d\xi'}.\]

Parameters:: end_to_end_length (numpy.ndarray) – The end-to-end length \(\xi\).
Returns:: The equilibrium probability density \(P_\mathrm{eq}\).
Return type:: numpy.ndarray

nondimensional_equilibrium_distribution(nondimensional_end_to_end_length_per_link)

The nondimensional equilibrium probability density of nondimensional end-to-end vectors per link as a function of the nondimensional end-to-end length per link,

\[\mathscr{P}_\mathrm{eq}(\gamma) = \frac{e^{-\Delta\vartheta(\gamma)}}{4\pi\int e^{-\Delta\vartheta(\gamma')} \,{\gamma'}{}^2 d\gamma'}.\]

Parameters:: nondimensional_end_to_end_length_per_link (numpy.ndarray) – The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).
Returns:: The nondimensional equilibrium probability density \(\mathscr{P}_\mathrm{eq}\equiv (N_b\ell_b)^3 P_\mathrm{eq}\).
Return type:: numpy.ndarray

equilibrium_radial_distribution(end_to_end_length)

The equilibrium probability density of end-to-end lengths as a function of the end-to-end length,

\[g_\mathrm{eq}(\xi) = 4\pi\xi^2 P_\mathrm{eq}(\xi),\]

which is calculated using the accurate analytic approximation provided by Becker et al.[1].

Parameters:: end_to_end_length (numpy.ndarray) – The end-to-end length \(\xi\).
Returns:: The equilibrium probability density \(g_\mathrm{eq}\).
Return type:: numpy.ndarray

nondimensional_equilibrium_radial_distribution(nondimensional_end_to_end_length_per_link)

The nondimensional equilibrium probability density of nondimensional end-to-end lengths per link as a function of the nondimensional end-to-end length per link,

\[\mathscr{g}_\mathrm{eq}(\gamma) = 4\pi\gamma^2 \mathscr{P}_\mathrm{eq}(\gamma).\]

Parameters:: nondimensional_end_to_end_length_per_link (numpy.ndarray) – The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).
Returns:: The nondimensional equilibrium probability density \(\mathscr{g}_\mathrm{eq}\equiv N_b\ell_b g_\mathrm{eq}\).
Return type:: numpy.ndarray

References