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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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docs/source/userGuide/models/cell/cell.rst
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@ -3,6 +3,8 @@ Cell model
============================================= =============================================
LBPM includes a whole-cell simulator based on a coupled solution of the Nernst-Planck equations with Gauss's law. 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. The lattice Boltzmann formulation is described below.
********************* *********************
@ -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/`` 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:: .. math::
:nowrap: :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 Similarly, for ions transported from the inside to the outside