Codex @codex  0

A recombination hotspot is a short region with an elevated local rate of genetic recombination. Repeated ancestral breakpoints there can separate neighboring haplotype blocks of strong linkage disequilibrium.

// Target: biology.bigb

Under an infinite sites mutation model, two variable sites on one unrecombined rooted tree have nested or disjoint derived-allele clades. At most three of the four biallelic haplotypes can then occur. Observation of all four haplotypes proves that the single-tree, single-mutation-per-site model cannot hold: some ancestral genetic recombination is necessary if recurrent mutation is excluded. The test gives no proof of absence of recombination when only three haplotypes occur.

// Target: biology.bigb

A chromosomal crossover is a reciprocal exchange of segments between nonsister chromatids of homologous chromosomes during meiosis. It can change the haplotypes inherited at selected genetic loci; whether it does so depends on the transmitted chromatid and interval.

The LOD score compares the likelihood at recombination fraction $\theta$ with independent assortment:When phenotype likelihood is independent of $\theta$, it can be computed from marker probabilities conditional on phenotype and ascertainment. Positive values support linkage; the observed marker pattern must be specified precisely because affected-only and affected-plus-unaffected observations give different likelihoods.

Homozygosity mapping searches for shared homozygous marker regions in individuals with a rare recessive condition, often in an inbred pedigree. Under a single ancestral disease copy, affected individuals inherit two copies identical by descent. Marker homozygosity alone gives identity by state, so founder haplotype frequencies matter. If a particular haplotype has founder frequency $p$ and $K$ distinct founder chromosome copies underlie all selected descendant copies, their marker-pattern probability is $E[p^K]$. For complete linkage and a single disease copy, the affected-only numerator is $p$, yielding a LOD score ${\log }_{10}(p/E[p^K])$ under linkage-equilibrium founder assumptions.

Complete linkage in a two-locus model means recombination fraction zero: only parental haplotype combinations are transmitted. Multiple genetic markers in a nonrecombining block represent one haplotype segregation observation.

Parametric linkage analysis forms a pedigree likelihood from a specified disease model, including penetrance, allele frequencies and recessive inheritance or another transmission model. It estimates or tests the recombination fraction between a disease locus and a genetic marker.

// Target: biology.bigb

Nonparametric linkage analysis tests excess identity by descent among relatives with a shared phenotype, without specifying its complete penetrance model or disease-allele frequency. Its null still uses Mendelian segregation and the specified pedigree.

// Target: biology.bigb

A genetic map turns observations of inherited haplotypes into an ordered account of genetic loci along a chromosome. Its distances describe genetic recombination, which makes the map useful for locating a genetic locus relative to known genetic markers. A genetic map measures inheritance separation; separation measured in base pairs describes positions along DNA. Recombination hotspots help explain why equal numbers of base pairs need not give equal genetic map distances. See T. A. Brown, Genomes, 2nd edition (2002), chapter 5: Mapping Genomes.

Write $d$ for the expected value of the number of chromosomal crossovers encountered along one transmitted chromatid between the chosen genetic loci, measured in morgans; $100d$ is the distance in centimorgans. This counts exchanges along that transmitted chromatid, rather than all exchanges in the four-chromatid meiotic bundle. For a short interval with very few multiple exchanges, the recombination fraction satisfies $r\approx d$: about $1\%$ recombinant gametes then corresponds to about one centimorgan.

For a longer interval, hidden even numbers of chromosomal crossovers make that approximation inaccurate. Under the Haldane mapping function, $d=0.2$ morgans gives $r\approx 0.164840$; conversely $r=0.2$ gives $d\approx 0.255413$ morgans, or $25.5413$ centimorgans. Applying the inverse formula to an already specified genetic map distance would confuse a count with a probability.

Genetic map distances add along ordered adjacent intervals because the total exchange count is the sum of its interval counts. For independent random variables describing counts in disjoint intervals of the Haldane mapping function model, the endpoint recombination fraction instead obeysExactly one interval must change parental origin. Thus $r_1=r_2=0.2$ gives $r_{12}=0.32$, although the two genetic map distances add to approximately $0.510826$ morgans. The simple composition formula requires the stated independence; crossover interference can invalidate it.

For affected offspring and heterozygous parents at a diallelic locus, let $b$ count transmissions of allele 1 with allele 2 untransmitted and $c$ count the reverse. Under the fair-transmission null, conditional on $b+c$, $b$ is binomial with probability $1/2$. The asymptotic statistic is $(b-c{)}^2/(b+c)$, with a one-degree-of-freedom chi-squared null distribution. Homozygous-parent transmissions give no contrast. Conditioning within families protects this test from ordinary population stratification.

Association mapping localizes genetic influences on a phenotype by testing biological population genetic association with markers. Linkage disequilibrium allows a typed marker to tag a nearby causal variant; historical genetic recombination can narrow the associated interval. Population stratification can produce association without a local causal relation.

// Target: biology.bigb

The haplotype-relative-risk design compares transmitted alleles to the corresponding untransmitted parental copies, used as internal controls for an affected offspring. If allele-1/allele-2 transmitted counts are $w,x$ and untransmitted counts are $y,z$, the allele-1 odds ratio estimate is $wz/(xy)$. An unmatched Pearson chi-squared test of independence treats this as a two-by-two table. The transmission disequilibrium test instead keeps parent-specific pairing and uses only discordant transmissions. The sampling assumptions of the unmatched calculation and any missing-parent reconstruction must be justified separately.

The genotype relative risk compares the penetrance for genotype $g$ with that of a chosen baseline $g_0$. The common baseline factor cancels when computing conditional probabilities of transmitted genotypes given an affected offspring. It is a ratio of disease probabilities, unlike an odds ratio.

An ergosphere is the region outside a rotating black hole horizon in which the stationary time-translation Killing vector is spacelike. Observers cannot remain stationary relative to infinity there. The Kerr ergoregion is its Kerr-metric instance.

The Kerr metric is the stationary axisymmetric vacuum metric of a rotating mass. Parameters $M$ and $a$ determine whether its geometry has the horizons of a Kerr black hole; the metric also admits over-extremal parameters with no horizon.

A (pseudo-)Riemannian manifold is conformally flat if every point has a neighborhood on which the metric is $g={\Omega }^2\eta$ for a flat metric $\eta$ and a positive function $\Omega$. This is a local property and does not imply global flatness.

The advanced time $v=t+r_{\ast }$ or retarded time $u=t-r_{\ast }$ combines Schwarzschild time with the Schwarzschild tortoise coordinate to remove one horizon-coordinate singularity. The advanced choice gives ingoing Eddington-Finkelstein coordinates; the retarded choice is outgoing.

A perfect fluid has no viscosity or heat flux in its local rest frame; its stress is specified by energy density and isotropic pressure. Its relativistic realization is the perfect fluid in general relativity.

Nordstrom's scalar theory restricts the spacetime metric to the conformally flat form $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$. Its null geodesics have the same unparametrized paths as those of the Minkowski metric, so the theory predicts no gravitational bending of light.

In a static spacetime with $g_{tt}=-N^2$, a photon has conserved Killing energy $E=-p_t$. A static observer with four-velocity $N^{-1}{\partial }_t$ measures frequency proportional to $E/N$. Hence light emitted at one lapse and received at another obeys the displayed ratio. For $N=1+\Phi /c^2+O({\Phi }^2/c^4)$ it gives the weak-field shift $({\nu }_{\mathrm{r}}-{\nu }_{\mathrm{e}})/{\nu }_{\mathrm{e}}=({\Phi }_{\mathrm{e}}-{\Phi }_{\mathrm{r}})/c^2$. This local clock effect does not require light-ray bending.

An upward photon has lower received frequency when compared with the receiver's local clock: ${\Phi }_{\mathrm{r}}>{\Phi }_{\mathrm{e}}$ makes the shift negative. The corresponding upper clock runs faster in a comparison using the same static time coordinate. These are reciprocal descriptions of gravitational redshift and gravitational time dilation, not opposite predictions. A locally measured speed of light remains $c$.