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Liisa Hirvonen edited Introduction.tex
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%\subsection{Radius calculation from MW}
%***This subsection needs to be rewritten.***
It is well known that increasing MW reduces the speed of diffusion across biological tissue,\cite{Maurice1977, Pitkanen2005, Geroski2001} and other studies have shown that the molecular radius is a better predictor of tissue penetration than MW.\cite{Ambati2002, Ambati2000a, Geroski2001, Bohrer1984, Ohlson2001, Venturoli2005} It is possible to estimate the radius of a protein from the MW. Erickson uses the fact that all proteins have approximately the same density, 1.37 g/cm$^3$, to calculate the protein volume from the MW.\cite{Erickson2009} Assuming a smooth spherical shape, this yields a minimum possible radius
\begin{equation}
R_{min}
\text{(nm)} = 0.066 \cdot \text{MW}^{1/3} \label{eq:Erickson}
\end{equation}
However, proteins have a rough surface, are often not perfectly spherical, and the ionic charge has an effect to the diffusion of a molecule in solution. The hydrodynamic radius $R_h$, defined as the radius of a hard sphere that diffuses at the same rate as that solute, takes these effects into account. The hydrodynamic radius is important in predicting transretinal penetration.\cite{Jackson2003} For example, Ambati et al.\ reported that the globular protein albumin, with a MW of 69~kDa and $R_h$ of 3.62~nm, had approximately twice the scleral permeability of a linear dextran of 40~kDa and $R_h$ of 4.5~nm.\cite{Ambati2000a}
Small-angle scattering studies using X-rays (SAXS) or neutrons (SANS) \cite{Svergun_2013} as well as dynamic light scattering (DLS) \cite{Pecora_1985, Hong_2009} and NMR techniques \cite{Wilkins1999} have been used for measuring $R_h$. Global analysis of hundreds of proteins has led to the definition of empirical relationships between $R_h$ and the number of amino acids $N$, related to the MW by \(N = \frac{\text{MW}}{110 \text{ Da}}\). Such formulas have been defined, for example, by Wilkins \cite{Wilkins1999}
\begin{equation}
R_h=4.75\cdot R_h (\text{\AA}) = 4.75\cdot N^{0.29} \label{eq:Wilkins}
\end{equation}
and Dill \cite{Dill2011}
\begin{equation}
R_h
(\text{\AA}) = 1.45\cdot(2.24\cdot N^{0.392}) = 3.248\cdot N^{0.392} \label{eq:Dill}
\end{equation}
%\subsection{Radius measurement from anisotropy}