Problems from part I touchups

Author: allardjun

ProblemSets_PartI/Math227C_P1.pdf

@@ -36,7 +36,7 @@
 \begin{document}
 
 
-\begin{enumerate}
+\begin{enumerate}[A.]
 
 %%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
 \item Two fair dice are rolled, one after the other.
@@ -47,11 +47,15 @@
 \end{enumerate}
 
 %%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
-\item Suppose a cell signaling network has five components. Each component acts independently with probability $p_i$, $i=1,2,3,4,5$. These components form a signaling pathway shown in the diagram below.
+\item Suppose a cell signaling network has five components. 
+Each component acts independently with probability $p_i$, $i=1,2,3,4,5$. 
+These components form a signaling pathway shown in the diagram below.
 \begin{figure}[h!]
 \centering\includegraphics[width=8cm]{figP12}
 \end{figure}
-The system is said to work if a signal originating at the left end of the diagram (the receptor) can reach the right end (the promotor), where it can pass through a component only if that component is working. For instance, if components 1 and 4 both work, then the system works. Component 3 can transduce a signal in either direction (some \emph{scaffold proteins} in cells provide such multivalent functions). What is the probability that the system works?
+The system is said to work if a signal originating at the left end of the diagram (the receptor) can reach the right end (the promotor), where it can pass through a component only if that component is working. 
+For instance, if components 1 and 4 both work, then the system works. Component 3 can transduce a signal in either direction (some \emph{scaffold proteins} in cells provide such multivalent functions). 
+What is the probability that the system works?
 
 \end{enumerate}
 

ProblemSets_PartI/Math227C_P1.tex

@@ -40,9 +40,12 @@
 
 %%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
 %\item % uncomments for multi-problem homeworks
-The DNA sequence of a gene determines the amino acid sequence of the protein it produces, with every 3 DNA basepairs coding for one protein amino acid. In eukaryotes, predicting the protein sequence from a gene sequence is not as simple as taking the gene sequence and splitting it into 3's, because of \emph{introns}, stretches of gene sequence that do not code for amino acids. (The regions of DNA in a gene that are not introns are called exons.)
+The DNA sequence of a gene determines the amino acid sequence of the protein it produces, with every 3 DNA basepairs coding for one protein amino acid.
+In eukaryotes, predicting the protein sequence from a gene sequence is not as simple as taking the gene sequence and splitting it into 3's, because of \emph{introns}, stretches of gene sequence that do not code for amino acids. 
+(The regions of DNA in a gene that are not introns are called exons.)
 
-A Markov chain model has been used to detect introns, assuming the 6 states shown in the diagram. The index set here does not represent time, but rather represents basepairs: Each increment is the transition from the previous basepair to the neighboring basepair.
+A Markov chain model has been used to detect introns, assuming the 6 states shown in the diagram. 
+The index set here does not represent time, but rather represents basepairs: Each increment is the transition from the previous basepair to the neighboring basepair.
 \begin{figure}[h!]
 \centering\includegraphics[width=8cm]{figP21}
 \end{figure}
@@ -58,8 +61,6 @@
 \item Under the assumptions of this model, what is the average length of a gene, including introns?
 \end{enumerate}
 
-
-
 % \end{enumerate} % uncomments for multi-problem homeworks
 
 \end{document}

ProblemSets_PartI/Math227C_P2.pdf

@@ -34,7 +34,7 @@
 \begin{document}
 
 
-\begin{enumerate}
+\begin{enumerate}[A.]
 
 %%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
 \item

ProblemSets_PartI/Math227C_P2.tex

@@ -36,19 +36,21 @@
 \begin{document}
 
 
-\begin{enumerate}
+\begin{enumerate}[A.]
 
 
 %%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
-\item Your advisor has scheduled two appointments with two graduate students, one at 1pm and the other at 1:30pm. The amounts of time that appointments last are independent exponential random variables with mean of 30 minutes. Assuming both graduate students arrive on time, find the expected amount of time that the 2nd grad student spends waiting outside the advisor's office (and/or Zoom meeting waiting room).
+\item Your advisor has scheduled two appointments with two graduate students, one at 1pm and the other at 1:30pm. 
+The amounts of time that appointments last are independent exponential random variables with mean of 30 minutes.
+Assuming both graduate students arrive on time, find the expected amount of time that the 2nd grad student spends waiting outside the advisor's office (and/or Zoom meeting waiting room).
 
 
 %%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
-\item The ``two-hit hypothesis'' was a model of cancer that assumes that two mutations are required for a normal cell to exhibit cancerous behavior.
+\item The ``two-hit hypothesis'' is a model of cancer that assumes that two mutations are required for a normal cell to exhibit cancerous behavior.
 Although the two-hit hypothesis is an oversimplification, it led indirectly to the discovery of tumor suppressor genes.
 % Although the two-hit hypothesis is an oversimplification, it led indirectly to the discovery of tumor suppressor genes \cite{tumor_suppressor}.
 
-Suppose there are two genes, Gene A and Gene B, each of which undergoes mutation at rate $\lambda$ (in units of, say, mutations per year).  
+Suppose there are two genes, Gene A and Gene B, each of which undergoes mutation at rate $\lambda$ (in units of mutations per year).  
 
 \begin{enumerate}[i.]
     \item How long, on average, until (either) one of the genes has mutated? 

ProblemSets_PartI/Math227C_P3.pdf

@@ -37,7 +37,7 @@
 
 The surface of a cell has a receptor where ligands become attached. 
 
-\begin{enumerate}
+\begin{enumerate}[A.]
 
 %%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%