\documentclass{../signatures}
\usepackage[utf8]{inputenc}
\labacronym{BMC}
\labtitle{Brownian Motion in Cells}

\begin{document}

\maketitle

\names

\prelab

\begin{enumerate}

    \item  What are the masses of the various microspheres (beads) you will be observing in the lab? How many molecules are in a single particle? What is the uncertainty in these numbers? These calculations help you get a sense of the size of your measurements. Data sheets for the microspheres are available in a black binder at the station.
    
    \item We assume that these beads in the fluid are in the non-inertial regime. Why is that true? What statistical assumption does that allow us to make? 

    \item Consider a 1-d random walk of N steps, where the probability of moving left equals the probability of moving right at each step. This allows us to define a random variable X  which represents the final displacement of the walker (its location after N steps). Calculate: $\langle X \rangle$ and $\langle X^2\rangle$. To do this, write $X=\sum_{1}^{N} S_i $, where $S_i$ is an indicator on the ith step (explicitly, $S_i=-1$ if the ith step was to the left, and $S_i=1$ if the ith step was to the right). Keep in mind the $S_i$ are independent when calculating $\langle X^2\rangle$.

    \item Learn the techniques to use pipettes. Show the GSI or faculty member how you use a pipette to deliver liquid (water) from one vial to another vial.
       \\[36pt]
\end{enumerate}

\prelabsignatures

\textbf{Questions to Complete on the First Day of Lab}

These questions can be answered after you read through the \href{http://experimentationlab.berkeley.edu/node/83}{\textbf{Simulating Brownian Motion}}. Make sure to copy and execute the scripts on the computer's MATLAB as you read so that you can understand the program structure and the different variable names. You have the freedom to choose other programming languages for these simulations, such as python. But so far, the example codes have been provided only in MATLAB.

In the first investigation, you will observe a minimum of two different-sized particles in at least four solvents with different viscosity. Do the following tasks for each of the experimental conditions you plan to observe: 

\begin{enumerate}

\item Simulate a random walk in Matlab or other programming language. (You should choose at least one particle with 1 $\mu m$ in size or larger and one smaller.) 
\item Plot the Displacement Squared as a function of time or number of steps.

\item Use your \textbf{simulated data} to calculate the diffusion coefficient, D in each case. Explain how you arrived at your answer. 

\item Calculate the uncertainty of D. How does it vary with the number of simulated data points? What additional sources of error (these are significant) will come into play? How will you account for them? The analysis script you are using could also be used on actual tracking data.
\\[36pt]
\end{enumerate}

\midlabsignatures{1}

\midlab

On day 2 of this lab, you should have completed the following. Show them to an instructor and ask for a signature.

\begin{enumerate}

    \item Using a slide with a combination of 10 $\mu m$ and 0.4 $\mu m$ polystyrene spheres, show how to set up K\"ohler illumination.
    
    \item How many nanometers per pixel are captured at 10x, 20x, and 40x magnification?
    
    \item Draw a diagram to show dark-field illumination. Explain how it is possible to see 40 nm objects with visible light (400-750 nm wavelengths).
    
\end{enumerate}

\midlabsignatures{2}

 By day 3 of this lab, you should have collected some particle tracks and made several movies. Show an instructor that you are able to make tracking work in the software, and present one of the particle tracks (csv file). \\
 \\[5pt]
 How can you extract diffusion coefficient D from the track? You can do this with either the BMC application or with the Matlab scripts. How close is this value to the theoretical and simulated values? Show to an instructor these calculations on at least one tracking dataset you obtain. \\
 \\[5pt]
 Show and explain your averaging and centroiding code. How do they work? 
\\[36pt]


\midlabsignatures{3}

\checkpointsection 

\begin{enumerate}

\item \checkp{Dark-Field Illumination}

\item \checkp{Data Analysis Additional Questions}

\item \checkp{Cellular Motion Additional Questions}

\end{enumerate}

\end{document}