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update cell do s

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James McClure 2023-01-17 09:02:55 -05:00
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10
docs/source/userGuide/models/cell/cell.rst
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@ -2,8 +2,10 @@
Cell model
=============================================
LBPM includes a whole-cell simulator based on a coupled solution of the Nernst-Planck equations with Gauss's law.
The lattice Boltzmann formulation is described below.
LBPM includes a whole-cell simulator based on a coupled solution of the Nernst-Planck equations with Gauss's law.
The resulting model is fully non-equilibrium, and can resolve the dynamics of how ions diffuse through the cellular
environment when subjected to complex membrane responses.
The lattice Boltzmann formulation is described below.
*********************
Nernst-Planck model
@ -209,13 +211,13 @@ interior and exterior. See the script ``NaCl-cell.py`` and input file ``NaCl.db`
Example input files for both cases are stored within the LBPM repository, located at ``example/SingleCell/``
The membrane simply prevents the diffusion of ions. All lattice links crossing the membrane are stored in a dedicated data structure so that transport is decoupled from the bulk regions. Suppose that site :math:`\mathbf{x}_{q\ell}` is inside the membrane and :math:`\mathbf{x}_{p\ell}` is outside the membrane. For each species :math:`k`, transport across each link :math:`\ell` is controlled by a pair of coefficients, :math:`\alpha^k_{\ell p}` and :math:`\alpha^k_{\ell q}`. Ions transported from the outside to the inside is
The membrane simply prevents the diffusion of ions. All lattice links crossing the membrane are stored in a dedicated data structure so that transport is decoupled from the bulk regions. Suppose that site :math:`\mathbf{x}_{q\ell}` is inside the membrane and :math:`\mathbf{x}_{p\ell}` is outside the membrane, with :math:`\mathbf{x}_{p \ell } = \mathbf{x}_{q\ell} + \bm{\xi}_q \Delta t`. For each species :math:`k`, transport across each link :math:`\ell` is controlled by a pair of coefficients, :math:`\alpha^k_{\ell p}` and :math:`\alpha^k_{\ell q}`. Ions transported from the outside to the inside is
.. math::
:nowrap:
$$
{ f_{q}^{k \prime} (\mathbf{x}_{q\ell}) \gets (1-\alpha^k_{\ell q}) f_{q}^{k} (\mathbf{x}_{q\ell}) + \alpha^k_{\ell p } f_{ p}^{k} (\mathbf{x}_{p\ell})}
{ f_{q}^{k \prime} (\mathbf{x}_{q \ell}) \gets (1-\alpha^k_{\ell q}) f_{q}^{k} (\mathbf{x}_{q\ell}) + \alpha^k_{\ell p } f_{ p}^{k} (\mathbf{x}_{p\ell})}
$$
Similarly, for ions transported from the inside to the outside