Tooluniverse Population Genetics

Population Genetics Analysis

MC Strategy: Population genetics MC questions often test whether you know a specific theorem or result. COMPUTE the answer first (use popgen_calculator.py or write Python), then match to options. Don't try to reason about which option "sounds right."

Analyze population-level genetic variation, allele frequencies, GWAS associations, clinical significance, and evolutionary constraints using ToolUniverse tools.

When to Use

Activate this skill when the user asks about:

• Allele frequencies across populations (gnomAD, 1000 Genomes)
• GWAS associations for diseases/traits
• Clinical variant interpretation (ClinVar, VEP)
• Gene-level constraint metrics (pLI, LOEUF, o/e ratios)
• Selection, drift, linkage disequilibrium, or population structure
• Variant annotation and functional consequences

LOOK UP, DON'T GUESS

Query gnomAD/1000Genomes/GWAS Catalog FIRST for allele frequencies and associations. Preferred: use the PopGen_hwe_test, PopGen_fst, PopGen_inbreeding, and PopGen_haplotype_count tools for HWE, Fst, inbreeding, and haplotype calculations. Fallback: run popgen_calculator.py directly. For theoretical problems (delta-q, drift, LD decay), apply the formulas in the Theoretical Reasoning section below.

Tool Quick Reference

Critical Gotchas

1. gnomAD variant_id: Format is "CHR-POS-REF-ALT" (no "chr" prefix). Always resolve rsIDs via gnomad_search_variants first.
2. gwas_search_associations: Takes disease/trait names ONLY. Gene names will fail. Use gwas_get_snps_for_gene for gene-based lookups.
3. gwas_search_snps: BROKEN (HTTP 500). Use gwas_get_snps_for_gene instead.
4. VEP/ClinVar responses: Format is variable (list, {data, metadata}, or {error}). Handle all three.

Workflow Patterns

Variant frequency: gnomad_search_variants -> gnomad_get_variant(dataset="gnomad_r4") -> MyVariant_query_variants (1000G pop breakdowns) -> EnsemblVEP_annotate_rsid

GWAS for disease: gwas_search_associations -> gwas_get_variants_for_trait -> gnomad_get_variant for top hits -> EuropePMC_search_articles

Gene characterization: gnomad_get_gene_constraints -> gwas_get_snps_for_gene -> ClinVar_search_variants -> PubMed_search_articles

Pathogenicity assessment: EnsemblVEP_annotate_rsid -> MyVariant_query_variants (CADD, ClinVar) -> gnomad_get_variant (frequency) -> RegulomeDB_query_variant (if non-coding)

Theoretical Reasoning (CRITICAL for computation problems)

These formulas are needed for quantitative population genetics problems. Work through step by step, showing intermediate values.

Allele Frequency Change Under Selection (delta-q)

For a recessive deleterious allele (fitness: AA=1, Aa=1, aa=1-s):

delta_q = -s * q^2 * p / (1 - s * q^2)

where p = freq(A), q = freq(a), s = selection coefficient.

For dominant deleterious (AA=1, Aa=1-s, aa=1-s):

delta_q = -s * q * p / (1 - s * q * (2 - q))

For heterozygote advantage (AA=1-s1, Aa=1, aa=1-s2):

equilibrium: q_hat = s1 / (s1 + s2)

Example: plug in s1 and s2 from the question; q_hat = s1/(s1+s2).

Selection against recessives is slow at low q because most a alleles hide in heterozygotes. Time to reduce q from q0 to qt: t ~ (1/qt - 1/q0) / s generations.

Genetic Drift in Small Populations

Variance in allele frequency per generation: Var(delta_p) = pq / (2Ne)

Probability of fixation of a new neutral mutation: 1/(2*Ne)

Time to fixation (given it fixes): ~4*Ne generations for neutral alleles

Heterozygosity decay: H_t = H_0 * (1 - 1/(2*Ne))^t

After t generations, fraction of heterozygosity lost ~ 1 - e^(-t/(2*Ne))

Effective population size (Ne) adjustments:

• Unequal sex ratio: Ne = 4NfNm / (Nf + Nm)
• Fluctuating size: Ne = harmonic mean of N across generations
• Bottleneck: dominated by the smallest generation size

Drift vs selection: Drift dominates when |s| < 1/(2*Ne). A variant with s=0.01 behaves neutrally in a population of Ne < 50.

Linkage Disequilibrium (LD) Decay

D = freq(AB) - freq(A)*freq(B), where A and B are alleles at two loci.

Decay with recombination: D_t = D_0 * (1 - r)^t, where r = recombination fraction, t = generations.

Half-life of LD: t_half = -ln(2) / ln(1-r) ~ 0.693/r generations (for small r).

r-squared (normalized LD): r^2 = D^2 / (pA * pa * pB * pb). Range 0-1.

Expected r^2 in finite population at equilibrium: E[r^2] = 1 / (1 + 4Ner) (for drift-recombination balance).

Practical implications:

• Tightly linked loci (r < 0.01): LD persists for hundreds of generations
• Loosely linked (r = 0.5, independent assortment): LD halves every generation
• GWAS tag SNPs work because LD extends over blocks; block size depends on Ne and recombination rate
• African populations have shorter LD blocks (larger historical Ne) -> need denser SNP arrays

Hardy-Weinberg Equilibrium

For alleles A (freq p) and a (freq q=1-p): expected genotypes AA=p^2, Aa=2pq, aa=q^2.

Chi-square test: df=1 (2 alleles). Preferred: use PopGen_hwe_test tool. Fallback: popgen_calculator.py --type hwe --AA N1 --Aa N2 --aa N3.

Causes of HWE departure: non-random mating, selection, migration, drift, genotyping error. Excess homozygotes -> inbreeding or population structure (Wahlund effect). Excess heterozygotes -> overdominant selection or negative assortative mating.

Heritability

• H^2 (broad-sense) = V_G / V_P; h^2 (narrow-sense) = V_A / V_P
• V_G includes ALL genetic variance: additive + dominance + epistasis. Trap: "broad-sense" is not just additive.
• Under HWE with two alleles (p, q): genotype frequencies are p^2, 2pq, q^2
• Phenotype frequency from genotype: sum(genotype_freq * penetrance) for each genotype class
• For quantitative traits: V_P = V_G + V_E (no covariance assumed)
• With dominance: assign genotypic values (e.g., AA=a, Aa=d, aa=-a), compute mean, then V_G from freq-weighted squared deviations
• PGS vs SNP-h² trap: PGS R² is NOT necessarily ≤ h²_SNP. With large GWAS, PGS can exceed SNP-h² by tagging rare causal variants through LD with common SNPs. The word "necessarily" makes this claim False. h²_SNP is estimated from common variants; PGS can capture additional variance.

Path Analysis (Causal Diagrams)

• Trace ALL paths from cause to effect through the diagram (direct + indirect)
• Each path's contribution = product of path coefficients along that path
• Total effect (correlation) = sum of contributions from all paths
• Indirect effects can mask (suppression) or inflate (confounding) the direct effect
• Unanalyzed correlations (double-headed arrows) count as valid path segments
• Never ignore indirect paths — the total is rarely just the direct arrow

Genetic Combinatorics (F2 crosses, haplotype counting)

For n SNPs between two inbred (homozygous) strains:

• F1 is heterozygous at all n loci
• F2 distinct haplotypes = 2^n (each SNP contributes parental A or B allele)
• F2 distinct diploid genotypes = 3^n (AA, AB, BB at each locus)
• F2 unique chromosomes (distinct haplotypes) = 2^n (e.g., 5 SNPs → 2^5 = 32; but subtract the 2 parental haplotypes if "novel" is asked → 30)
• ALWAYS write and run Python code (python3 -c "...") for these counts. Never enumerate by hand.
• For specimens/counting from field data: parse the data into a structure and compute programmatically.

Mutation-Selection Balance

Equilibrium frequency of a deleterious allele:

• Recessive lethal: q_hat = sqrt(mu/s) ~ sqrt(mu) when s=1
• Dominant lethal: q_hat = mu/s
• Example: mu=1e-5, s=1 (recessive lethal) -> q_hat = 0.003 (carrier freq ~ 0.006)

F-statistics and Population Structure

• Fis: Inbreeding within subpopulations (heterozygote deficit within demes)
• Fst: Differentiation between subpopulations. Fst = Var(p) / (p_bar * q_bar)
• Fit: Total inbreeding. (1-Fit) = (1-Fis)(1-Fst)
• Fst interpretation: <0.05 little, 0.05-0.15 moderate, 0.15-0.25 great, >0.25 very great differentiation
• Preferred: use PopGen_fst tool. Fallback: popgen_calculator.py --type fst --p1 X --p2 Y --n1 N1 --n2 N2

Mendelian Genetics Reasoning Framework

For any genetics cross problem, follow these steps IN ORDER. Do not skip steps.

Step 1: Identify genes, locations, and allele relationships

• List every gene involved in the cross
• Determine chromosomal location: autosomal vs X-linked (X-linked genes show different inheritance in males vs females)
• Determine allele relationships: dominant/recessive, codominant, incomplete dominance
• Note any epistasis, suppressor, or modifier interactions between genes

Step 2: Write parental genotypes explicitly

• Use standard notation (e.g., Aa Bb for autosomal; X^w X^+ for X-linked)
• For X-linked genes, males are hemizygous (X^w Y), not homozygous
• If parental genotypes are not given, deduce them from phenotypes and pedigree context

Step 3: Draw Punnett square(s) for each gene

• For multi-gene crosses, handle each gene independently (if unlinked) then combine
• For linked genes, use recombination frequency to adjust gamete ratios
• For X-linked genes, remember that fathers pass X to all daughters and Y to all sons

Step 4: Calculate expected phenotypic ratios

• Multiply independent gene ratios (e.g., 3:1 x 3:1 = 9:3:3:1)
• For X-linked: calculate male and female ratios separately, then combine or report separately as required

Step 5: Verify ratios sum to 1.0

• Convert all ratios to fractions and confirm they sum to 1
• If they don't sum to 1, there is an error in the Punnett square or gamete calculation

Step 6: Apply phenotype modification rules AFTER computing genotypic ratios

• For epistasis: first compute the full genotypic ratios (e.g., 9:3:3:1), then collapse genotype classes that produce the same phenotype
• For suppressor genes: a suppressor homozygote (su/su) restores wild-type in an otherwise mutant background. Apply suppression AFTER determining which individuals carry the mutant allele
• Example: 9 A_B_ : 3 A_bb : 3 aaB_ : 1 aabb with recessive epistasis (aa masks B) becomes 9:3:4

E. coli Hfr Mapping Framework

For bacterial conjugation and Hfr mapping problems:

Core Principles

• In Hfr x F- crosses, the Hfr chromosome is transferred linearly starting from the origin of transfer (oriT)
• Gene transfer order = chromosomal order from the origin
• Early markers (entering first) are closest to the origin of transfer
• Late markers (entering last) are farthest from the origin

Interrupted Mating Experiments

• Genes that appear in recombinants at earlier time points are closer to oriT
• The time of entry gives the order and approximate distance between genes
• Recombinants require integration by homologous recombination (double crossover)

Recombination Frequency Between Markers

• KEY TRAP: Highest recombination frequency occurs between markers that are FARTHEST APART on the transferred segment
• This is because more time elapses between entry of distant markers, providing more opportunity for recombination events between them
• Conversely, markers that enter close together in time show LOW recombination between them
• Do NOT confuse "highest recombination frequency" with "first markers to enter" -- these are opposite concepts

Ordering Markers from Hfr Data

1. Use time-of-entry data to establish gene order relative to oriT
2. Use recombination frequency data between pairs of selected markers to confirm/refine order
3. Multiple Hfr strains with different origins can be used to build a circular map

MCQ Elimination Strategy for Genetics

General MCQ Protocol

1. ALWAYS evaluate ALL options before choosing an answer
2. Never select the first option that seems correct -- there may be a better or more precise answer
3. Read the question stem carefully for qualifiers: "MOST likely", "LEAST likely", "NOT true", "ALWAYS", "NEVER"

"Which is NOT true" Questions

• Evaluate EACH statement independently as True or False
• Mark each option with T or F before selecting
• The answer is the statement marked F
• Double-check: verify the "false" statement is genuinely false, not just misleadingly worded

"Which mechanism" Questions

• Test each proposed mechanism against ALL observations given in the question
• A correct mechanism must explain every observation, not just some
• Eliminate mechanisms that contradict even one observation

Specific Traps to Watch For

• Subfunctionalization vs neofunctionalization: Subfunctionalization = partitioning of EXISTING ancestral functions between duplicates (both copies needed to perform original function). Neofunctionalization = one copy acquires a genuinely NEW function not present in the ancestor
• Copy-neutral LOH: Caused by mitotic recombination (segmental, affects part of a chromosome), NOT uniparental disomy (UPD, which is whole-chromosome). The question may try to conflate these
• Penetrance vs expressivity: Penetrance = fraction of individuals with genotype who show ANY phenotype. Expressivity = degree/severity of phenotype among those who show it. These are distinct concepts
• Complementation vs recombination: Complementation = two mutations in DIFFERENT genes restore wild-type in trans. Recombination = exchange between two mutations in the SAME or different genes. Complementation is tested in F1 (heterozygote); recombination is tested in progeny

Common Genetics Reasoning Traps

These are specific patterns that have caused reasoning failures in hard genetics questions. Review before answering genetics MCQs.

Suppressor Genetics

• A suppressor mutation, when homozygous, restores wild-type phenotype in an otherwise mutant background
• In F2 crosses involving both the original mutation and an autosomal recessive suppressor:
  • Treat as a dihybrid cross — the primary mutation and the suppressor segregate independently
  • Only 1/4 of F2 are homozygous for the suppressor
  • The suppressor only acts in individuals that are also homozygous for the primary mutation
  • Use a Punnett square to enumerate all genotypic classes, then apply the suppression rule to determine phenotypes

Non-disjunction (Bridges' Experiments)

• Bridges used non-disjunction to prove the chromosome theory of inheritance
• X0 males arise from female meiosis non-disjunction events
• Meiosis I non-disjunction: both X chromosomes go to one pole -> XX egg + O egg (nullo-X)
• Meiosis II non-disjunction: sister chromatids fail to separate -> XX egg from one secondary oocyte
• The classic Bridges result: exceptional white-eyed females (X^w X^w) and red-eyed males (from nullo-X eggs + Y sperm = X0, but these are typically sterile)
• Key distinction: know which type of non-disjunction (MI vs MII) produces which specific gamete types

GWAS LD Blocks

• SNPs WITHIN the same LD block are correlated and can inflate false positive associations (one causal SNP drags along non-causal tag SNPs)
• SNPs ACROSS different LD blocks are largely independent and do NOT create misleading cross-locus associations
• LD block structure varies by population (shorter in African populations due to larger historical Ne)
• Fine-mapping within an LD block is needed to distinguish the causal variant from hitchhiking tag SNPs

Gene Retention After Whole-Genome Duplication

• Neofunctionalization: One copy acquires a NEW function -> most commonly cited reason for gene RETENTION after duplication (preserves both copies because each is now essential)
• Subfunctionalization: Ancestral functions are PARTITIONED between copies -> explains DIVERGENCE of duplicate copies, but both copies must be retained to maintain the full ancestral function
• Dosage balance: Some genes are retained in duplicate to maintain stoichiometric balance in protein complexes
• Trap: Questions may ask "what explains retention" vs "what explains divergence" -- these have different best answers
• For retention: neofunctionalization (new function makes both copies essential)
• For divergence of expression/function: subfunctionalization (partitioning of ancestral roles)

Advanced Genetics Traps v2

PGS vs Heritability: "Necessarily True" Logic

For "necessarily true" questions about PGS and heritability: a statement is necessarily true only if it holds when V_D=0 AND when V_D=V_G. Test the extremes.

Path Diagram Sign Assignment Protocol

Do NOT guess path signs from general knowledge. Signs may differ from well-known systems. Follow this protocol:

1. Establish reference direction: What varies? What is increasing?
2. For each path X→Y: Ask ONLY "when X increases, does Y increase (+) or decrease (-)?"
3. Use the question's experimental context (knockout/control comparisons, provided data) to determine signs — not intuition
4. Expect negative paths: Path diagrams test your ability to identify negative relationships. All-positive is almost always wrong. Direct residual paths (e) often have opposite sign from expectation.

Chi-Square: "Most Likely to Reject" Protocol

Compute chi-square from the expected ratio given in the question. Compare to chi-square-critical at df = (number of phenotype classes - 1). Pick the answer choice with the highest chi-square, but also check which pattern is biologically diagnostic of the alternative hypothesis.

LD and Misleading GWAS Associations

LD block boundaries at recombination hotspots are a source of GWAS false localization — strong signal in the block does not guarantee the causal variant is in the block.

Low-Frequency Allele Detection

Duplex sequencing (unique molecular identifiers + double-strand consensus) detects alleles at 0.01% frequency — far below standard NGS even at 80X depth. Simply increasing read depth does NOT help for ultra-rare variants because the Illumina error rate (~0.1%) masks variants rarer than ~1% regardless of depth. Error correction methods (UMIs, duplex consensus) are needed to distinguish true rare variants from sequencing errors.

Bundled Computation Script

Script: skills/tooluniverse-population-genetics/scripts/popgen_calculator.py

Preferred: Use ToolUniverse tools (via MCP/SDK) instead of the script when possible:

• PopGen_hwe_test tool -- HWE chi-square test. Fallback: popgen_calculator.py --type hwe
• PopGen_fst tool -- Weir-Cockerham Fst. Fallback: popgen_calculator.py --type fst
• PopGen_inbreeding tool -- Inbreeding coefficient from pedigree. Fallback: popgen_calculator.py --type inbreeding
• PopGen_haplotype_count tool -- Expected haplotype diversity. Fallback: popgen_calculator.py --type haplotypes

Fallback script modes (all require --type):

• hwe: --AA N --Aa N --aa N -- chi-square HWE test with p-value
• fst: --p1 F --p2 F --n1 N --n2 N -- Weir-Cockerham Fst
• inbreeding: --pedigree TYPE --generations G -- F from pedigree (self, full-sib, half-sib, first-cousin, etc.)
• haplotypes: --snps N --generations G --recomb_rate R -- expected haplotype diversity

Key Concepts

• MAF: Minor allele frequency. Common: >5%. Rare: <1%. Ultra-rare: <0.01%.
• pLI: P(LoF intolerant). >0.9 = haploinsufficient gene.
• LOEUF: LoF o/e upper fraction. <0.35 = highly constrained.
• CADD PHRED: >=10 top 10%, >=20 top 1%, >=30 top 0.1% most deleterious.
• Genome-wide significance: GWAS p < 5e-8 (Bonferroni for ~1M independent tests).
• Effect size: OR > 1 = risk allele, < 1 = protective. Beta > 0 = increases trait.

Evidence Grading

• T1: ClinVar pathogenic/likely pathogenic, FDA pharmacogenomics
• T2: gnomAD frequencies, GTEx eQTLs, GWAS genome-wide significant
• T3: CADD/SIFT/PolyPhen predictions, RegulomeDB, constraint metrics
• T4: VEP consequence terms, dbSNP annotations, literature mentions