\graphicspath{ {./images/} }

\subsection{Biology background}
To understand the function and behavior of ion channels, several points need to
be discussed. The background
comes from a wide disciplinary range, including biology, chemistry,
thermodynamics, electricity, probability theory, and differential equations.
We will only be able to present a brief overview here. 
For more information, the reader is referred to Hille's extensive book,
\emph{Ion Channels of Excitable Membranes} \cite{hille-2001}, and Keener and
Sneyd's work on the applied mathematics of cellular physiology
\cite{keener-2009}.

Before discussing the biology of cells and ion channels any further, a quick
discussion regarding ionic solutions and the
\emph{electrochemical gradient} is needed. 
As any good student should know, ions are atoms or molecules that have a
positive or negative charge due
to differing numbers of protons and electrons. Specifically, anions have more electrons than
protons, so they have a negative charge. Conversely, cations
are ``missing'' electrons, and therefore have a positive charge. 

Ions arise from a variety of natural processes, including the dissolution of salts
in water. For example, when table salt, or
more officially, sodium chloride
($NaCl$) dissolves in water, sodium cations with one positive charge ($Na^+$)
and chloride anions with one negative charge ($Cl^-$) become dissociated and
free to move around independently. Similarly, the salts potassium chloride
($KCl$) and calcium
chloride ($CaCl_2$) dissolve into solutions of free $K^+$, $Cl^-$, and $Ca^{2+}$
ions. 

Biological cells are divided into two main groups: prokaryotic cells which
include bacteria, and eukaryotic cells such as plant and animal cells.
Prokaryotic and eukaryotic cells are both enclosed in a selectively permeable
biological membrane known as the cell (or plasma) membrane. Eukaryotic
cells also have a
number of internal membranes enclosing their nucleus, mitochondria, and
several other organelles. Prokaryotes on the other hand do not have a nucleus or
any other membrane-bound organelles. 

The cell membrane of both prokaryotes and
eukaryotes, as well as the membranes of certain organelles all contain ion
channels. From this point on, however, the focus will be on the cell membrane
of eukaryotic cells. 

The cell membrane separates the intracellular contents from the
extracellular environment and is composed of a thin double layer of lipids about
7.5 $nM$ thick \cite{keener-2009}. 
%Figure
%\ref{fig:cell_membrane} provides a view of a typical cell membrane. %%
%
%%%%% Cell membrane  %%%%
%\begin{figure}
%\begin{center}
% 
%\fbox{\includegraphics[width=2.5in]
%{cell_membrane.eps}}
%    \caption{Representation of the cell membrane showing the lipid bilayer,
%integral and peripheral proteins, and carbohydrates. From Wikipedia \cite{wiki-cellmembrane}.}
%    \label{fig:cell_membrane}
%\end{center}
%\end{figure}

An important characteristic of the cell membrane is that it is selectively
permeable, meaning that it allows certain molecules to pass though in either
direction in controlled quantities. For example, waste products are allowed to
leave the cell, and certain sugars, acids, and other molecules can enter.
Also affected by the membrane's selective permeability are ions, such as
$Na^+$ (sodium), $K^+$ (potassium), $Cl^-$ (chloride), and $Ca^{2+}$ (calcium).

The transportation of molecules across the membrane can either be passive
(requiring no net energy), or active (requires energy, such as ATP). An
important active transport mechanism is the
$Na^+/K^+-ATPase$, which is also called the sodium-potassium pump for the
sake of readability. This pump is a trans-membrane protein that uses the
energy stored in ATP to pump $Na^+$ out of the cell and $K^+$ into the cell
\cite{keener-2009}. In addition to pumps, integral proteins called exchangers
also facilitate active transport across the cell membrane. The $Na^+-Ca^{2+}$
exchanger uses the energy gained from allowing $Na^+$ entry into the cell to
remove $Ca^{2+}$. One may question why allowing $Na^+$ into the cell provides
energy, while removing $Ca^{2+}$ requires energy. The reason this is so is due
to the electrochemical gradients arising from the ionic concentration levels
that the cell maintains through active transport. 

At the cell's resting potential, the concentration of $Na^+$ is much higher
outside the cell than inside the cell, so $Na^+$ wants to enter because the
electrochemical gradient is driving it into the cell.
$Ca^{2+}$ also a higher extracellular
concentration, so it takes energy to remove it from the cell because it must be
moved up, or \emph{against} its electrochemical gradient.


The last transport mechanism that deserves mention are ion channels,
trans-membrane proteins which facilitate the diffusion of select ion species
down their electrochemical gradient. Ion channels are therefore
referred to as passive transport mechanisms. As a simplification, an individual channel 
can be thought of as either open, or closed. 
In the open configuration, channels allow ions to pass
through at speeds of $~10^8$ ions per second, near diffusion speed
\cite{capener-2002}.

From the perspective of the whole cell, the large number ion channels together
can be said to affect the \emph{permeability} of the membrane to a certain ion
species.  

In summary, the cell membrane uses ion channels, pumps, and exchangers to
maintain a certain concentration gradient for a number of ion species. The
membrane's permeability to an ion species depends on the types of ion
channels in that cell type, as well as their current states (open, or closed).


\subsection{Modeling ion channels}
The modeling of ion channels dates back over 100 years
\cite{levitt-1999}. As
is with many mathematical models, the motivations driving the development of ion
channel models include:
(1) striving to infer fundamental knowledge of the underlying physical
processes,
(2) matching experimental data to the model's predictions,
(3) using the model to predict behaviors which are difficult to observe, and
(4) developing common terminology and knowledge among researchers.


The interpretation of ion channels models during the pre-molecular era provided
the primary source of information about channel structure \cite{levitt-1999}.
However, current technology such as molecular biology and x-ray diffusion are
beginning to prove atomic-resolution detail of bacterial as well as some
mammalian channels, though the fraction of channels with known structure
remains low \cite{capener-2002, levitt-1999}. 

Hodgkin and Huxley's landmark model describing the ionic mechanisms behind
the action potential of a squid giant axon provided the foundation for many
modern practices in electrophysiology and ion channel modeling
\cite{hodgkin-1952, keener-2009}. Their model describes the voltage-dependent
and time-dependent behavior of $Na^+$ and $K^+$ conductances using a coupled set
of ordinary differential equations. While the Hodgkin-Huxley
model is still used today due to its simplicity and low number of parameters, it
does exhibit several shortcomings. These include, for example, a lack
of connectivity between activating and inactivating gates in the $Na^+$ channel,
as well as the premise that the inactivation gate can only close after the
activation
gate opens \cite{gurkiewicz-2007}.

{\bf Markov models:}
Markov models for ion channels are an extension of the Hodgkin-Huxley formalism, 
and are good for modeling single channel data, gating currents, and drug
interactions. Unlike Hodgkin-Huxley models, Markov models show the state
dependence of activation and inactivation \cite{fink-2009}. 

{\bf Theory: }
Ion channels can be modeled as continuous time Markov processes, 
where a simplification of the channels functional
physical shape is represented by states in the Markov chain. A model can have
several open, closed, deactivated, and inactivated states, as well as states
which represent the binding of ligands. The rates connecting the various
states are time independent kinetic rates which can be constant, or dependent on
the membrane voltage, or intra- and extra-cellular ionic concentrations.

{\bf Criticism:}
Markov models are not without critics. In a one page perspective
titled \emph{Are rate constants constant?} Jones challenges the community to
examine the benefits of time-dependent rate constants over Markov models
\cite{jones-2006}. Jones cites the findings by Uebachs \cite{uebachs-2006} who
reports that the rate of recovery from inactivation in a type of
$Ca^{+2}$ channel depends on the length of depolarization, which goes against
the basic principle of chemical kinetics that rate constants remain constant in
a constant environment \cite{jones-2006}.

Although Markov models are memoryless by definition, the time-dependent behavior
described
by Uebachs can be reproduced by having a long chain of open or closed states
\cite{jones-2006, liebovitch-1989}. Nevertheless, Jones concludes that
time-dependent rate constants as used by fractal channel models, would provide a
more explicit memory than multistate Markov models.  

{\bf Fractal Models:}
Liebovitch has been the main proponent of fractal ion channel models for over
15 years \cite{liebovitch-1989, lowen-1999, liebovitch-2001}. One of his main arguments questioning the use
of Markov models is founded in the body of evidence depicting ion channel
proteins as complex structures with a continuum of states, exhibiting large and
small motions over varying time scales. Fractal models as proposed by Liebovitch
contain a large number of conformational states with kinetic rates connecting
them. The rates, however, are linked and not independent as in Markov models.
This has the effect of giving the channel ``memory'' \cite{liebovitch-1989}. 

An additional concern raised by Liebovitch is the use of exponential rate
constants. He provides a friendly reminder to his colleagues
that the fitting of experimental data to sums of exponentials is famously
ill-conditioned \cite{liebovitch-1989, acton-1970}. As confirmation he provides
a quote from a numerical methods text which states that those who try to
determine the parameters of such equations from experimental data ``must be
spanked or counseled. At the very least, keep them from obstructing Progress and
the computer!'' \cite{acton-1970}. 

In their defense, other researchers have argued that Markov models with
exponential rate constants and a small number of states fit the experimental
data better than fractal models \cite{mcmanus-1988}. Nevertheless, fractal
models continue as the number one challenger to Markov models.







