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Question 2 MCQ · 3.0 marks
In a double-stranded DNA molecule, a researcher determines that **22%** of all bases (counting both
strands together) are adenine (A).
Recall that DNA obeys strict complementary base pairing: adenine (A) always pairs with thymine
(T), and cytosine (C) always pairs with guanine (G). Using this constraint, what percentage of the
total bases in the double-stranded molecule are guanine (G)?
6406537043084.
22%
6406537043085.
28%
6406537043086.
11%
6406537043087.
50%
Published solution
**1. Apply Chargaff's base-pairing rules:**
In a double-stranded DNA molecule, Adenine (`A`) always pairs with Thymine (`T`) and Guanine (`G`) always pairs with Cytosine (`C`). Therefore:
- \( \%\text{A} = \%\text{T} = 22\% \)
- Together, \( \%\text{A} + \%\text{T} = 22\% + 22\% = 44\% \).
**2. Calculate the percentage of Guanine (`G`):**
The remaining bases are equally split between `G` and `C`:
\[
\%\text{G} + \%\text{C} = 100\% - 44\% = 56\% \implies \%\text{G} = \frac{56\%}{2} = 28\%
\]
**Answer: B — 28%.**
A: **Incorrect:** `22%` is the percentage of Thymine (`T`), which pairs with Adenine (`A`).
B: **Correct:** \( (100\% - 2 \times 22\%) / 2 = 56\% / 2 = 28\% \).
C: **Incorrect:** Divides `22%` by 2 instead of subtracting `A + T` from `100%`.
D: **Incorrect:** `50%` would leave `0%` for `A` and `T`.
Consider a hypothetical organism with exactly **4 genes**. Each gene can be in one of three states:
OFF, ON at low expression, or ON at high expression. Assuming every combination of gene states
defines a distinct cell type, how many theoretically possible cell types can this organism produce?
**1. Set up the combinatorial count:**
- Number of independent genes: \( n = 4 \)
- Number of possible expression states per gene: \( k = 3 \) (`OFF`, `ON low`, `ON high`)
**2. Calculate the total number of distinct state combinations:**
\[
\text{Number of cell types} = 3^4 = 81
\]
**Answer: B — 81.**
A: **Incorrect:** \( 4^3 = 64 \) swaps the base (states per gene = 3) and the exponent (number of genes = 4).
B: **Correct:** With 3 states for each of the 4 genes, there are \( 3^4 = 81 \) possible combinations.
C: **Incorrect:** \( 4 \times 3 = 12 \) multiplies instead of exponentiating.
D: **Incorrect:** \( 2^4 = 16 \) only accounts for 2 states per gene instead of 3.
A researcher models a gene regulatory network among 4 genes (A, B, C, D) as a directed graph,
where an edge from gene
to gene
means "gene
regulates gene
" (regardless of whether the
regulation is activation or inhibition). The observed regulatory relationships are:
• Gene A activates Gene B
• Gene B activates Gene C
• Gene C inhibits Gene A (forming a negative feedback loop)
• Gene B activates Gene D
• Gene C activates Gene D
How many directed edges does this graph have, and how many genes have an in-degree of
exactly 1?
6406537043092.
5 edges; 3 genes have in-degree 1.
6406537043093.
4 edges; 2 genes have in-degree 1.
6406537043094.
5 edges; 2 genes have in-degree 1.
6406537043095.
4 edges; 4 genes have in-degree 1.
Published solution
**1. List the directed edges in the Gene Regulatory Network:**
1. \( A \to B \) (A activates B)
2. \( B \to C \) (B activates C)
3. \( C \to A \) (C inhibits A)
4. \( B \to D \) (B activates D)
5. \( C \to D \) (C activates D)
This gives a total of **5 directed edges**.
**2. Compute the in-degree of each gene:**
- **Gene A:** incoming edge from `C` \( \implies \text{in-deg}(A) = 1 \)
- **Gene B:** incoming edge from `A` \( \implies \text{in-deg}(B) = 1 \)
- **Gene C:** incoming edge from `B` \( \implies \text{in-deg}(C) = 1 \)
- **Gene D:** incoming edges from `B` and `C` \( \implies \text{in-deg}(D) = 2 \)
Thus, **3 genes** (`A, B, C`) have an in-degree of exactly 1.
**Answer: A — 5 edges; 3 genes have in-degree 1.**
A: **Correct:** There are 5 directed edges in total, and genes A, B, and C each have in-degree 1 (while D has in-degree 2).
B: **Incorrect:** Undercounts both total edges and in-degree-1 nodes.
C: **Incorrect:** Three genes (`A, B, C`), not two, have in-degree 1.
D: **Incorrect:** There are 5 edges, and Gene D has in-degree 2.
**1. Analyze the 3-D Dynamic Programming state \( B(v, C, r) \):**
Here \( B(v, C, r) \) is the maximum weight of a path ending at vertex \( v \) that uses the color subset \( C \) and contains **exactly \( r \) proteins from the special set \( T \)**.
**2. Consider whether the endpoint \( v \) belongs to \( T \):**
When extending a path from predecessor \( u \) to \( v \) (removing \( v \)'s color \( c(v) \) from \( C \) to get \( C \setminus \{c(v)\} \)):
- **Case 1 (\( v \in T \)):** Vertex \( v \) itself contributes 1 protein from \( T \), so the prefix path ending at \( u \) must contain \( r - 1 \) proteins from \( T \):
\[
\max_{u} \big\{ B(u, C \setminus \{c(v)\}, r - 1) + w(u, v) \big\}
\]
- **Case 2 (\( v \notin T \)):** Vertex \( v \) contributes 0 proteins from \( T \), so the prefix path ending at \( u \) must already contain \( r \) proteins from \( T \):
\[
\max_{u} \big\{ B(u, C \setminus \{c(v)\}, r) + w(u, v) \big\}
\]
This piecewise recurrence matches **Option A**.
**Answer: A.**
A: **Correct:** Decrements the count \( r \) to \( r - 1 \) when \( v \in T \) and keeps \( r \) unchanged when \( v \notin T \).
B: **Incorrect:** Reverses the two cases (decrementing \( r \) when \( v \notin T \)).
C: **Incorrect:** Fails to distinguish whether \( v \in T \) or \( v \notin T \).
D: **Incorrect:** Increments \( r \) instead of decrementing \( r \) when looking up the prefix subproblem.
In a protein--protein interaction (PPI) study, a logistic regression model is trained to estimate the
probability that two proteins truly interact, using three features computed for each protein pair
:
•
: number of independent experimental studies that observed the interaction,
•
: Pearson correlation of gene expression between the genes encoding
and
,
•
: small-world clustering coefficient (fraction of shared interaction partners in the PPI network).
The trained model is:
with learned parameters
,
,
,
.
A protein pair
has feature values
,;
,;
. Compute the predicted
interaction probability
.
Enter the value correct to 2 decimal places.
Published solution
**1. Compute the linear logit \( z = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 \):**
Substituting the given parameters and feature values (where \( z = 2 \ln 2 \)):
\[
z = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 = 2 \ln 2 = \ln 4
\]
**2. Evaluate the logistic sigmoid function \( P = \frac{1}{1 + e^{-z}} \):**
\[
P = \frac{1}{1 + e^{-\ln 4}} = \frac{1}{1 + 1/4} = \frac{1}{1.25} = 0.80
\]
**Answer: 0.80.**
To discover signaling pathways in a yeast PPI network, researchers first estimate the probability of
true interaction for each protein pair. They train a logistic regression model of the form:
where
is the number of independent high-throughput experiments (such as yeast two-hybrid
or affinity purification--mass spectrometry studies) that detected the interaction, and
is the
Pearson correlation of gene expression between the two proteins' encoding genes.
For a particular protein pair, the gene expression correlation is
. What is the minimum
integer value of
(the number of experimental studies) required so that the model considers the
interaction more likely than not (i.e.,
)?
Enter the value as a single integer.
Published solution
**1. Set up the threshold condition for \( P(y = 1 \mid x_1, x_2) \ge 0.5 \):**
In logistic regression, \( P(y = 1 \mid \mathbf{x}) = \frac{1}{1 + e^{-z}} \ge 0.5 \) if and only if the linear predictor \( z \ge 0 \).
**2. Substitute the model coefficients and \( x_2 = 0.5 \):**
From the given formula \( z = -4 + x_1 + 2 x_2 \) with \( x_2 = 0.5 \):
\[
z = -4 + x_1 + 2(0.5) = x_1 - 3 \ge 0 \implies x_1 \ge 3
\]
Thus, the minimum integer number of experimental studies is **3**.
**Answer: 3.**
After estimating interaction probabilities using logistic regression, the edge weight in a PPI graph
is defined as
, so that path scores (sums of edge weights) correspond to log-
probabilities.
Consider a small PPI network with 5 proteins:
. The edge weights are:
Protein
is a membrane receptor and protein
is a transcription factor. A signaling pathway is
modeled as a simple path (no repeated proteins) from
to
.
What is the **score of the highest-scoring simple path** from
to
?
Enter the value as an integer.
Published solution
**1. List all simple paths from receptor \( A \) to transcription factor \( E \) and sum their edge weights:**
From the given edge weights in the 5-protein network:
- **Path 1 (\( A - B - D - E \)):** \( w(A,B) + w(B,D) + w(D,E) = (-1) + (-1) + (-2) = -4 \)
- **Path 2 (\( A - C - D - E \)):** \( (-3) + (-2) + (-2) = -7 \)
- **Path 3 (\( A - B - C - D - E \)):** \( (-1) + (-2) + (-2) + (-2) = -7 \)
**2. Select the maximum (least negative) path score:**
The highest-scoring simple path is \( A \to B \to D \to E \) with score **-4**.
**Answer: -4.**
receives information about a pathogen and transmits the information to gene
via
the following edges.
•
to
with edge probability
.
•
to
with edge probability
.
•
to
with edge probability
.
•
to
with edge probability
.
In a random coloring of the nodes of the underlying GRN network using
colors, what is the
probability that this path is colorful? (Random coloring refers to coloring each node in the graph
with one of the
colors uniformly and independently at random.)
Round up the answer to 2 decimal points.
Published solution
**1. Count the nodes in the path and the number of colors:**
- The path \( R \to P_1 \to P_2 \to T \to G \) has 4 edges and therefore **5 nodes**.
- Each node is colored independently and uniformly at random using one of **7 colors**.
**2. Compute the probability that all 5 nodes receive distinct colors:**
- Total colorings of the 5 path nodes: \( 7^5 = 16,807 \)
- Favorable colorings (all 5 nodes get different colors from the 7 available colors): \( 7 \times 6 \times 5 \times 4 \times 3 = 2,520 \)
\[
P(\text{colorful}) = \frac{7 \times 6 \times 5 \times 4 \times 3}{7^5} = \frac{2,520}{16,807} \approx 0.149937 \approx 0.15
\]
**Answer: 0.15.**
You travel to an alien world where the DNAs are composed of a whopping
nucleotides, namely,
and are tasked with reading the gene
using two different machines.
Let the noisy set of
-mer composition of
generated by the first machine is represented as
and the noisy set of
-mer composition of
generated by the second machine
is represented as
.
Note that, both of these compositions are generated by the machines, and thus, may contain
missing and erroneous
-mers.
Now, due to errors in the
-mers read by both machines, the sequence constructed from de Bruijn
graph using
is BDACDDG and the sequence constructed from the overlap
graph using
is FBFDEDDG.
Answer the given subquestions with respect to the same.
What is the maximum possible out-degree of any node in the de Bruijn graph constructed using
?
Enter the value as a single integer.
Published solution
**1. Analyze the maximum possible out-degree of a node in a de Bruijn graph:**
In a de Bruijn graph over an alphabet \( \Sigma \) of size \( |\Sigma| = 8 \) (`A, B, C, D, E, F, G, H`), each node represents a \( (k-1) \)-mer \( u \). An outgoing edge from \( u \) corresponds to a distinct \( k \)-mer whose \( (k-1) \)-prefix is \( u \), formed by appending one nucleotide \( \sigma \in \Sigma \) to the right of \( u \).
**2. Conclude:**
Since there are **8** distinct nucleotides in the alien alphabet (and \( C_1 \) is a set of \( k \)-mers), a node can have at most **8** distinct outgoing edges.
**Answer: 8.**
You travel to an alien world where the DNAs are composed of a whopping
nucleotides, namely,
and are tasked with reading the gene
using two different machines.
Let the noisy set of
-mer composition of
generated by the first machine is represented as
and the noisy set of
-mer composition of
generated by the second machine
is represented as
.
Note that, both of these compositions are generated by the machines, and thus, may contain
missing and erroneous
-mers.
Now, due to errors in the
-mers read by both machines, the sequence constructed from de Bruijn
graph using
is BDACDDG and the sequence constructed from the overlap
graph using
is FBFDEDDG.
Answer the given subquestions with respect to the same.
We say a position has been read correctly, if the nucleotide at that position in the optimal global
alignment between the sequences are the same. Find the global alignment score with +1 scoring
for a match and 0 penalty for mismatches and gaps, and hence, determine how many positions of
the sequence have been read correctly.
Enter the value as a single integer.
Published solution
**1. Formulate the alignment problem as Longest Common Subsequence (LCS):**
With Match = \( +1 \) and Mismatch = Gap = \( 0 \), the optimal global alignment score equals the length of the **Longest Common Subsequence (LCS)** of:
- Sequence 1: `BDACDDG` (length 7)
- Sequence 2: `FBFDEDDG` (length 8)
**2. Find the LCS:**
Aligning `B`, `D`, and the suffix `DDG` gives the common subsequence **`BDDDG`** of length **5**:
- `- B - D A C D D G`
- `F B F D E - D D G`
No common subsequence of length 6 exists. Thus, **5** positions have been read correctly.
**Answer: 5.**
You travel to an alien world where the DNAs are composed of a whopping
nucleotides, namely,
and are tasked with reading the gene
using two different machines.
Let the noisy set of
-mer composition of
generated by the first machine is represented as
and the noisy set of
-mer composition of
generated by the second machine
is represented as
.
Note that, both of these compositions are generated by the machines, and thus, may contain
missing and erroneous
-mers.
Now, due to errors in the
-mers read by both machines, the sequence constructed from de Bruijn
graph using
is BDACDDG and the sequence constructed from the overlap
graph using
is FBFDEDDG.
Answer the given subquestions with respect to the same.
How many
-mers were read by the second machine?
Enter the value as a single integer.
Published solution
**1. Relate the reconstructed sequence length to the number of \( k \)-mers:**
- The sequence reconstructed from the second machine's overlap graph is `FBFDEDDG`, which has length \( L = 8 \).
- The second machine reads \( k \)-mers of length \( k = 6 \) (6-mers).
**2. Count the number of 6-mers in a sequence of length 8:**
\[
N = L - k + 1 = 8 - 6 + 1 = 3
\]
(Specifically: `FBFDED`, `BFDEDD`, and `FDEDDG`).
**Answer: 3.**
**1. Extract the 3-mer composition of `TTATTCGC` (length \( L = 8 \)):**
The number of 3-mer edges is \( m = 8 - 3 + 1 = 6 \):
`TTA`, `TAT`, `ATT`, `TTC`, `TCG`, `CGC`.
**2. Find all unique 2-mer prefixes and suffixes (nodes \( n \)):**
The 2-mers are:
- `TT`, `TA`, `AT`, `TC`, `CG`, `GC`
All **6** of these 2-mers (`{TT, TA, AT, TC, CG, GC}`) are distinct! Therefore:
- Number of nodes: \( n = 6 \)
- Number of edges: \( m = 6 \)
**Answer: A — \( n = 6,\ m = 6 \).**
A: **Correct:** The sequence `TTATTCGC` has 6 edges (3-mers) and 6 unique 2-mer nodes (`TT, TA, AT, TC, CG, GC`).
B: **Incorrect:** All six 2-mers are distinct, so \( n = 6 \), not 5.
C: **Incorrect:** A string of length 8 has \( 8 - 3 + 1 = 6 \) 3-mer edges, not 7.
D: **Incorrect:** Neither \( n \) nor \( m \) is 7.
**1. Extract the \( (3, 1) \)-mers (paired 3-mers at distance \( d = 1 \)) from `TTATTCGC` (length \( L = 8 \)):**
Each \( (k, d) \)-mer spans \( 2k + d = 2(3) + 1 = 7 \) characters. Thus the number of paired edges is:
\[
m = L - (2k + d) + 1 = 8 - 7 + 1 = 2 \text{ edges}
\]
Specifically:
1. Indices 1..7 (`TTATTCG`): `(TTA | TCG)` with prefix `(TT | TC)` and suffix `(TA | CG)`.
2. Indices 2..8 (`TATTCGC`): `(TAT | CGC)` with prefix `(TA | CG)` and suffix `(AT | GC)`.
**2. Count the unique paired 2-mer nodes \( n \):**
The unique nodes are `(TT | TC)`, `(TA | CG)`, and `(AT | GC)`, which gives **\( n = 3 \) nodes** and **\( m = 2 \) edges**.
**Answer: C — \( n = 3,\ m = 2 \).**
A: **Incorrect:** Does not match the span \( 2k + d = 7 \) on a string of length 8.
B: **Incorrect:** Swaps the node and edge counts.
C: **Correct:** Two \( (3,1) \)-mers (`TTA|TCG` and `TAT|CGC`) form \( m = 2 \) edges connecting \( n = 3 \) unique paired 2-mer nodes.
D: **Incorrect:** Overcounts both nodes and edges.
CCTGAATGC.
Based on the above data, answer the given subquestions.
We would like to align
and
using the following score matrix (S).
What is the alignment score for the global alignment between the following two strings:
Enter the value as a single integer.
Published solution
**1. Score each column of the given 9-column global alignment using matrix \( S \):**
The given alignment between \( S_1 = \text{CAAT-GATC} \) and \( S_2 = \text{CCTGAATGC} \) has 9 columns:
1. `(C, C)` (match)
2. `(A, C)` (mismatch)
3. `(A, T)` (mismatch)
4. `(T, G)` (mismatch)
5. `(-, A)` (gap)
6. `(G, A)` (mismatch)
7. `(A, T)` (mismatch)
8. `(T, G)` (mismatch)
9. `(C, C)` (match)
**2. Sum the column scores from \( S \):**
Looking up each pair in the provided scoring matrix \( S \) and summing across all 9 columns gives a total alignment score of **-9**.
**Answer: -9.**
**1. Evaluate each candidate alignment under the scoring scheme (Match = +1, Mismatch = -1, Gap = -2):**
For \( S_1 = \text{CAATGATC} \) (length 8) and \( S_2 = \text{CCTGAATGC} \) (length 9):
- **Option C** aligns 6 matching bases (`C, T, G, A, T, C` or `C, A, A, T, G, C`), 1 mismatch, and 2 gaps (or 7 matches, 0 mismatches, 3 gaps: \( 7(+1) + 3(-2) = +1 \)), achieving the highest total score among the four options.
**Answer: C.**
A: **Incorrect:** Introduces extra mismatches and gaps, yielding a lower total score than Option C.
B: **Incorrect:** Has fewer matches than Option C.
C: **Correct:** Maximizes the number of matched nucleotides while minimizing gap and mismatch penalties.
D: **Incorrect:** Suboptimal placement of gaps reduces the alignment score.
