\graphicspath{ {./images/} }

Several example models of varying complexity have been created and simulated
using ModFossa. 
%A description of the each model is provided, along with a listing of
%the Python source code used define the model. 
In this section we will present one such example.  Visual results are provided
in the form of plots generated by the code. Finally, an examination of
ModFossa's computational performance is provided.

\subsection{Model Description}
In \cite{angermann-2006}, Angermann et al. attempt to determine how phosphatase
activity influences calcium-activated chloride channels in rabbit pulmonary
artery myocytes (smooth muscle cells). 

To simulate the behavior of the test and control groups, Angermann et al.
created a Markov chain kinetic model based on the work by Kuruma and Hartzell
\cite{kuruma-2000}. Slightly different parameters were chosen for the test and
control groups in an attempt to make clear the difference in behavior between
the two. The model, shown in Figure \ref{fig:angermann_model}, consists of four
closed states and three open states. Calcium binding occurs in the closed states
with rates directly proportional to $[Ca]$. State $C_1$ is representative of the
channel's state when no $Ca^{2+}$ ions are bound to the receptor sites, while
state $C_4$ represents the channel with all three $Ca^{2+}$ binding sites
occupied. The closed states with at least one binding site occupied can
transition into open states with constant channel opening rates. Closed states
with more occupied binding sites have higher values for the channel opening
rates, giving the model its concentration-dependent behavior. Voltage
dependent channel closing rates take the model from open to closed states. 

In \cite{angermann-2006}, the behavior of the channel in the presence of
AMP-PNP was simulated by setting the gating variables of all three of the
channel's open states to 1. Conversely, to simulate the channel in the presence
of ATP, only channel $O_1$ was assigned a gating variable of 1, while the rest
of the open states were given a value of zero. This reproduces the
blocking effect by phosphorylation that the authors hypothesize as a possible
explanation for the inhibition of $I_{ClCa}$ while the channel is subjected to
ATP. In addition to reducing the channel gating variables to model the effect of
ATP, the authors also increased the magnitude of the channel closing rates and
shifted the voltage dependence towards more positive potentials.

By altering the Markov model's parameters in an insightful manner, Angermann et
al. were able to simulate the behavior of $Ca^{2+}$-activated $Cl^-$ currents
in the presence of both AMP-PNP and ATP. 



\subsection{Model implementation in ModFossa}

The calcium activated chloride channel Markov model shown in Figure
\ref{fig:angermann_model} was implemented in ModFossa. The
parameter values were chosen to match the behavior of $I_{ClCa}$ in the
presence of ATP, and were modified slightly from Thibeault
et al. \cite{thibeault-2011}.

  %%%% Markov Model %%%%
\begin{figure} 
\begin{center}
\fbox{\includegraphics[width=3.0in]
{angermann_model.eps}}
    \caption{Angermann CaCl Markov model. The model has four closed states and
three open states. State C2 represents the channel with a single $Ca^{2+}$
binding site occupied, while state C4 represents the channel with all three
binding sites occupied. Note that the model can only transition to an open
state in the presence of at least one bound $Ca^{2+}$.}
    \label{fig:angermann_model}
\end{center}
\end{figure}
\subsection{ModFossa output}

A  chord conductance plot was generated using ModFossa (Figure
\ref{fig:angermann_ca_g_vs_v}). This plot shows channel conductance as a
function of voltage, for 6 different concentration values. A chord
conductance plot of Angermann et. al's measured experimental data is shown in
Figure \ref{fig:angermann_g_vs_v_atp_experimental}. As shown, the simulated data
(Figure \ref{fig:angermann_ca_g_vs_v}) match the experimental data (Figure
\ref{fig:angermann_g_vs_v_atp_experimental}) well.

%%%% G vs V %%%%
\begin{figure}
\begin{center}
 
\fbox{\includegraphics[width=3.0in]
{angermann_ca_g_vs_v.eps}}
    \caption{Simulated Angermann CaCl conductance at varying concentrations, as
a function of membrane voltage. This figure was
generated using ModFossa's \emph{plotGvsV} method.}
    \label{fig:angermann_ca_g_vs_v}
\end{center}
\end{figure}

%%%% G vs V Experimental%%%%
\begin{figure}
\begin{center}
 
\fbox{\includegraphics[width=3.0in]
{angermann_g_vs_v_atp_experimental.eps}}
    \caption{Measured data from \cite{angermann-2006} showing conductance as a
function of voltage and concentrations. The data were fit to a Boltzmann
function using the least squares method.}
    \label{fig:angermann_g_vs_v_atp_experimental}
\end{center}
\end{figure}


The runtimes of ModFossa versus a similar Matlab implementation are provided in
Figure \ref{fig:runtime_graph}. ModFossa is approximately 17 times faster.

%%%% Runtime %%%%
\begin{figure}
\begin{center}
 
\fbox{\includegraphics[width=2.9in]{runtime.eps}}
    \caption{ModFossa runtime versus a Matlab implementation for the example model. 
The x-axis corresponds to the number of experiment sweeps performed at each data
point.}
    \label{fig:runtime_graph}
\end{center}
\end{figure}


