# Satya Arjunan/sandbox

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(New page: This is an email I wrote to Prof Gleb Oshanin, putting it here just for my own records: Dear Prof Oshanin, Thank you for sending your paper about the determination of the reaction probab...)
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This is an email I wrote to Prof Gleb Oshanin, putting it here just for my
own records:

Dear Prof Oshanin,

Thank you for sending your paper about the determination of the reaction
probability p from the integral reaction rate, Q_N. I also greatly
appreciate your willingness to answer my questions. I apologize if I am
bothering you with trivial questions. I do not have any physics faculty
members here and I am doing my research independently. Most of my work is
related to biology. I hope to hear your opinion about a Monte-Carlo
simulation method that I have developed. I have attached the method in the
pdf file because it would be easier for you to view the equations.

With reference to the method, I would really be happy to hear your

1. Is the method acceptable, i.e., correct, to reproduce both the reaction
and diffusion dynamics of particles in a liquid using a lattice?
2. If it is not correct, what do you think I should do?
3. Would it be possible for me to apply the lattice-based method described
in your paper to reproduce the reaction-diffusion dynamics of particles in
a liquid?
4. Do you know of any other lattice-based methods that you think is
appropriate?
5. Can I relate this method with the Collins and Kimball mean-field
approach since they say the local reaction rate at a point should be
equivalent to the diffusion of two particles into the point?

I really appreciate it even if you can give very short answers to these
questions. I am very good at programming (especially in C and C++

FYI, I am currently evaluating the survival probability of the target
particles in my method using the Equation 9 in the attached paper,
"Coarse-grained molecular simulation of diffusion and reaction kinetics in
a crowded virtual cytoplasm". I hope that it will be correct.

Best regards,
Satya Arjunan

Contents of the attached method in latex source:
\documentclass[a4paper,english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\usepackage[version=3]{mhchem}
\usepackage{kerkis}
\usepackage{biophysj}
\usepackage{booktabs,array,url,graphicx,courier,xspace}
\usepackage[final]{microtype}
\usepackage{upgreek,ifpdf,listings,float,caption,longtable}

\usepackage{babel}

\begin{document}
Dear Prof Oshanin, my aim is to develop a lattice-based Monte-Carlo
simulation method that accurately reproduces the reaction-diffusion
dynamics of two mobile particles, $A$ and $B$, in a liquid with
a volume, $V\mathrm{m}^3$. The overall reaction in $V$ is given
by $A + B \xrightarrow{k_a} C$, where $k_a$ has the unit,
$\mathrm{number}^{-1}.\mathrm{m}^3.\mathrm{s}^{-1}$.
There are $N_A$, $N_B$ and $N_C$ number of $A$, $B$ and $C$
particles respectively. I am using a 3D lattice with a lattice spacing
of $a$, such that $a=R_A=R_B$, where $R_A$ and $R_B$ are the diameters
of the particle $A$ and $B$ respectively. To make all the particles
diffuse, i.e., randomly walk to a neighbor site in each time step
$\Delta t$, I let

\begin{align}
\Delta t=\frac{a^2}{6D}
\end{align}where $D$ is the diffusion coefficient of $A$ and $B$. When the
scavenger particle $A$ meets a target particle $B$ at its destination
site, they can react with a probability $p$. I determined $p$ from
the macroscopic rate constant $k_a$ with the following approach:
Say in the volume $V$ there are $N_S$ sites. At each simulation
step, the probability of finding a target particle $B$ by a particle
$A$ at its destination site is

\begin{align}
p_1=\frac{N_B}{N_S}
\end{align}In a time step, the average number of target particles found by
all
scavenger particles is

\begin{align}
p_2&=N_A p_1\\
&=\frac{N_A N_B}{N_S}
\end{align}Out of the $p2$ target particles found in a time step, if some
of
them react and $\Delta N_C$ particles are formed, then

\begin{align}
p&=\frac{\Delta N_C}{p2}\\
&=\frac{\Delta N_C N_S}{N_A N_B} \label{p}
\end{align}In the liquid, we know from the law of mass action that

\begin{align}
\frac{d[C]}{dt}=k_a[A][B]
\end{align}In a very small $\Delta t$ within the volume V

\begin{align}
\Delta N_C = \frac{k_a N_A N_B}{V} \Delta t \label{deltaNc}
\end{align}Substituting \eqref{deltaNc} in \eqref{p}

\begin{align}
p&=\frac{k_a N_S}{V}\Delta t
\end{align}
\end{document}