Biology & Genetics — CheriMathLab

🧬 Key Concepts & Laws

Central Dogma

$$\text{DNA} \rightarrow \text{RNA} \rightarrow \text{Protein}$$ Information flows from nucleic acids to proteins via transcription and translation.

Mendelian Genetics

Law of Segregation and Independent Assortment. Phenotypic ratios: $3:1$ (monohybrid), $9:3:3:1$ (dihybrid).

Hardy-Weinberg Equilibrium

$$p^2 + 2pq + q^2 = 1 \quad\text{and}\quad p + q = 1$$ Allele frequencies remain constant in the absence of evolution.

DNA Replication

Semi-conservative replication. Helicase unwinds, primase adds primers, DNA polymerase III synthesizes $5' \rightarrow 3'$, ligase seals gaps.

Natural Selection

Differential survival and reproduction based on fitness. Drives adaptation through heritable variation in populations.

Gene Expression

Regulated at transcriptional, post-transcriptional, translational, and post-translational levels. Operons, enhancers, silencers.

CRISPR-Cas9

Guide RNA directs Cas9 nuclease to specific DNA sequences for precise gene editing — knock-outs, insertions, or corrections.

Population Genetics

Genetic drift, gene flow, mutation, and selection alter allele frequencies. Bottleneck and founder effects reduce diversity.

✎ Problems

1 Easy

In a cross between two heterozygous pea plants ($Rr \times Rr$), where $R$ (round) is dominant over $r$ (wrinkled), predict the genotypic and phenotypic ratios of the offspring. If 200 offspring are produced, how many would you expect to be wrinkled?

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Draw a Punnett square: $RR$, $Rr$, $Rr$, $rr$. Genotypic ratio = $1:2:1$. Phenotypic ratio = $3:1$ (round:wrinkled). Wrinkled = $rr = \tfrac{1}{4}$ of 200.

2 Easy

A DNA template strand reads $3'\text{-TACGGAATCCTG-}5'$. Write the complementary mRNA sequence, then translate it into an amino acid sequence using the standard genetic code. Identify the start and stop signals.

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mRNA is complementary and antiparallel to the template strand: $\text{A}\rightarrow\text{U}$, $\text{T}\rightarrow\text{A}$, $\text{C}\rightarrow\text{G}$, $\text{G}\rightarrow\text{C}$. Read $5' \rightarrow 3'$. The first codon AUG codes for Met (start). Read in triplets to find the amino acid chain.

3 Medium

Quantitative Genetics: In a population, 16% of individuals show the recessive phenotype ($aa$) for a trait. Assuming Hardy-Weinberg equilibrium, calculate the frequencies of the dominant allele ($p$), recessive allele ($q$), homozygous dominant ($AA$), and heterozygous ($Aa$) individuals.

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If $aa = 16\%$, then $q^2 = 0.16$, so $q = 0.4$. Then $p = 1 - q = 0.6$. $AA = p^2 = 0.36$ (36%). $Aa = 2pq = 0.48$ (48%). Verify: $0.36 + 0.48 + 0.16 = 1.00$.

4 Medium

In a dihybrid cross ($AaBb \times AaBb$), where both traits show complete dominance, what fraction of offspring will be: (a) dominant for both traits, (b) recessive for both traits, (c) dominant for $A$ and recessive for $B$? Show the expected ratio.

Show Hint

By independent assortment, treat each gene separately: $A\_ = \tfrac{3}{4}$, $aa = \tfrac{1}{4}$, $B\_ = \tfrac{3}{4}$, $bb = \tfrac{1}{4}$. Multiply probabilities: (a) $\tfrac{3}{4} \times \tfrac{3}{4} = \tfrac{9}{16}$, (b) $\tfrac{1}{4} \times \tfrac{1}{4} = \tfrac{1}{16}$, (c) $\tfrac{3}{4} \times \tfrac{1}{4} = \tfrac{3}{16}$. The classic $9:3:3:1$ ratio.

5 Medium

A human gene has $3{,}000$ base pairs. During DNA replication, the error rate is approximately $1$ per $10^9$ base pairs (after proofreading). If the human genome has $6.4 \times 10^9$ base pairs, how many mutations would you expect per cell division? Per generation (assuming ~30 cell divisions in the germline)?

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Mutations per division $= \text{genome size} \times \text{error rate} = 6.4 \times 10^9 / 10^9$. Per generation: multiply by the number of germline divisions. This gives a sense of the raw mutation rate that drives evolution.

6 Medium

Quantitative Genetics: Sickle-cell disease (HbS allele) is maintained at a frequency of $q = 0.1$ in a West African population due to heterozygote advantage against malaria. If the fitness of $AA = 0.88$, $AS = 1.0$, and $SS = 0.14$, verify that $q = 0.1$ is an equilibrium frequency using the selection balance equation.

Show Hint

At equilibrium with heterozygote advantage: $\hat{q} = \dfrac{1 - w_{11}}{(1 - w_{11}) + (1 - w_{22})} = \dfrac{s_1}{s_1 + s_2}$ where $s_1 = 1 - 0.88 = 0.12$ and $s_2 = 1 - 0.14 = 0.86$. Check if $\hat{q} \approx \dfrac{0.12}{0.12 + 0.86}$.

7 Hard

A population of 500 organisms undergoes a genetic bottleneck, reducing to 20 individuals. If the original allele frequency for a neutral allele was $p = 0.3$, what is the expected allele frequency after the bottleneck? What is the standard deviation of the allele frequency due to genetic drift? What is the probability the allele is lost entirely?

Show Hint

Expected frequency remains $p = 0.3$ (drift is random). Standard deviation $= \sqrt{\dfrac{p(1-p)}{2N}} = \sqrt{\dfrac{0.3 \times 0.7}{40}}$. For complete loss: model as binomial — probability all 40 alleles ($2N = 40$) are non-$p$ = $(1 - p)^{2N}$.

8 Hard

A bacterial population doubles every 20 minutes. Starting from a single cell, how many bacteria will there be after 8 hours? If each bacterium occupies $1\;\mu\text{m}^3$, what total volume would they occupy? Compare this to everyday objects. Why doesn't this actually happen in nature?

Show Hint

Number of generations $= 480/20 = 24$. $N = 2^{24} = 16{,}777{,}216$ bacteria. Volume $= N \times 1\;\mu\text{m}^3$. Convert to mL ($1$ mL $= 10^{12}\;\mu\text{m}^3$). In nature, resources run out — logistic growth applies.

9 Hard

CRISPR Design Challenge: You want to knock out a gene by introducing a frameshift mutation using CRISPR-Cas9. The target sequence near the start codon is 5'-ATGCCTGGATGGAACCTTGA-3', and the PAM site (NGG) is underlined. Design a 20-nt guide RNA. If Cas9 cuts 3 bp upstream of the PAM and causes a 1-bp insertion, how does this affect the reading frame?

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The guide RNA is complementary to the strand opposite the PAM. Cas9 cuts between positions 3 and 4 upstream of the PAM (TGG). A 1-bp insertion shifts all downstream codons by $+1$, causing a frameshift that produces a premature stop codon and nonfunctional protein.

10 Hard

Quantitative Genetics: Flower color in snapdragons shows incomplete dominance: $RR$ = red, $Rr$ = pink, $rr$ = white. In a population with 120 red, 240 pink, and 140 white flowers, test whether the population is in Hardy-Weinberg equilibrium using a chi-squared test ($\chi^2$).

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Count alleles: $p = \dfrac{2 \times 120 + 240}{2 \times 500} = \dfrac{480}{1000} = 0.48$. $q = 0.52$. Expected: $RR = p^2 \times 500 = 115.2$, $Rr = 2pq \times 500 = 249.6$, $rr = q^2 \times 500 = 135.2$. Calculate $\chi^2 = \sum \dfrac{(O - E)^2}{E}$ with 1 degree of freedom. Compare to critical value $3.84$ ($\alpha = 0.05$).

11 Advanced

The heritability ($h^2$) of human height is approximately $0.8$. In a population with mean height $170$ cm, parents with average height $180$ cm are selected. Predict the mean height of their offspring using the breeder's equation: $R = h^2 \times S$, where $R$ is the response to selection and $S$ is the selection differential.

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Selection differential $S = \text{parental mean} - \text{population mean} = 180 - 170 = 10$ cm. Response $R = h^2 \times S = 0.8 \times 10 = 8$ cm. Predicted offspring mean $= 170 + 8 = 178$ cm. Note: this assumes additive genetic variance dominates.

12 Advanced

Deep Thought: The human genome contains approximately $20{,}000$ protein-coding genes, but over 98% of the genome is non-coding. A typical gene produces an average of 3 splice variants. Calculate the theoretical number of distinct proteins the human genome can produce. Then discuss: if each protein can be modified post-translationally in at least 5 ways, what is the effective "proteome" size? How does this complexity arise from a seemingly simple genome?

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Distinct mRNAs $\approx 20{,}000 \times 3 = 60{,}000$. With 5 PTMs each: effective proteome $\approx 60{,}000 \times 5 = 300{,}000+$ distinct functional proteins. The combinatorial explosion of alternative splicing plus PTMs explains how organisms with similar gene counts (humans vs. nematodes) achieve vastly different complexity.