# Satya Arjunan/sandbox

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 comments about the following questions:

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++ languages). If there is anything I can do to help you please let me know.

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}