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For a specified characteristic, complete dominance means that the heterozygote has the same phenotype as one of the two homozygotes. It does not mean that the dominant allele is more common, more advantageous, or transmitted more often. Penetrance specifies the conditional probability of an expressed phenotype given a genotype and is a separate modeling assumption.

Let $q_g=P(g\mid g_f,g_m)$ be the Mendelian segregation probability, and assume the child's disease status depends on the parental genotypes only through its own genotype, with penetrance proportional to $R_g$. Bayes' theorem gives the displayed conditional law. For a diallelic parent pair there are four equally likely labelled transmission outcomes. If an unordered genotype occurs in $m_g$ of them, $q_g=m_g/4$, so its probability is $m_g{R}_g/{\sum }_h{m}_h{R}_h$. Dropping multiplicities would, for example, misweight a heterozygous child of two heterozygotes.

For a binary disease trait $Y$ with genotype $G$, the genetic variance of penetrance is the between-genotype term in the law of total variance. The averaged within-genotype term is $\mathbb{E}[q_G(1-q_G)]$. Under multiplicative diallelic penetrance, the genetic term is $p^2({\beta }^2-{\alpha }^4)$. This decomposition attributes variance to the modeled locus, not automatically to all hereditary causes.

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.

Assign each factor exactly once to a containing clique and multiply the factors there into ${\psi }_i$. The displayed sum-product belief propagation message sums out variables of $C_i$ outside the separator $S_{ij}=C_i\cap C_j$. By the running intersection property, eliminating all variables in the subtree on the $i$ side creates precisely this function of the separator. An inward pass computes the normalizing constant at the root; an outward pass produces marginal distributions. For pedigree likelihoods, observed genotypes and penetrance factors simply multiply the relevant potentials.

In ordinary diploid meiosis, the two copies at a genetic locus separate: each gamete receives one copy. With fair transmission, a heterozygote $Aa$ produces gametes carrying $A$ or $a$ with probability $1/2$ each. This is a transmission mechanism conditional on a parental genotype, rather than a statement about how common either allele is in a population.

For the one-locus cross $Aa\times Aa$, assume independent parental transmissions, no new mutation, and no differences in fertilization or survival between the resulting genotypes. Multiplying the two gamete probabilities gives four equally likely ordered outcomes: $AA,Aa,aA,aa$, each with probability $1/4$. Combining the two orders of the heterozygote givesOnly under the additional ideal model of complete dominance of $A$, complete penetrance, and no environmental changes to the scored characteristic do $AA$ and $Aa$ share the dominant phenotype. Its probability is then $0.25+0.50=0.75$. The inheritance calculation alone does not determine a phenotype or predict a generic disease risk.

These are probabilities, not a requirement that every four offspring contain one $AA$, two $Aa$ and one $aa$. For 100 independent offspring under the ideal model, the expected values of the three counts are $25,50,25$; the $aa$ count has a binomial distribution with standard deviation $\sqrt{100(0.25)(0.75)}\approx 4.33$. Each new transmission keeps the same conditional probabilities given the parental genotypes.

In his paper published in 1866, Mendel reported 5,474 round and 1,850 wrinkled pea seeds: $5474/7324\approx 0.7474$ round, close to the ideal $0.75$. Following descendants also distinguished two inherited classes within the round phenotype, illustrating why genotype cannot always be read from appearance.^[ref] At several genetic loci, transmission patterns additionally depend on genetic linkage and genetic recombination.

With Hardy-Weinberg equilibrium, let $G$ count copies of the risk allele. The penetrance $p{\theta }^G$ must lie in $[0,1]$ for every possible genotype. Writing $\alpha =1-\varphi +\varphi \theta$ and $\beta =1-\varphi +\varphi {\theta }^2$ gives $\mathbb{E}(q_G)=p{\alpha }^2$ and $\mathbb{E}(q_G^2)=p^2{\beta }^2$.

For ordered alleles $I,J$ independently drawn with probabilities ${\pi }_i$, suppose disease penetrance is $c{\psi }_i{\psi }_j$. Conditional on disease, the allele draws remain independent with probabilities ${\pi }_i^{\ast }={\pi }_i{\psi }_i/{\sum }_u{\pi }_u{\psi }_u$. This follows by normalizing the product $c{\pi }_i{\pi }_j{\psi }_i{\psi }_j$. Unordered homozygote probabilities are $({\pi }_i^{\ast }{)}^2$ and heterozygote probabilities are $2{\pi }_i^{\ast }{\pi }_j^{\ast }$. Unaffected controls do not generally share this factorization: their genotype weights contain $1-c{\psi }_i{\psi }_j$.

// 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

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

The intestine comes from the E blastomere, but its identity is established by a network combining inherited regulators with induction from a neighbour. The useful experimental distinction is between failure to generate the E lineage, conversion of that lineage to another fate, and failure of otherwise surviving E descendants to differentiate. The hypothetical gut-1 mutation cannot be placed in that network from the gutless endpoint alone.

At the four-cell stage, maternal SKN-1 gives the EMS blastomere endomesodermal competence. EMS divides into anterior MS and posterior E. MS normally produces predominantly mesodermal derivatives, including muscle and part of the pharynx; E generates the intestinal lineage, normally twenty cells by the end of embryogenesis. This ancestry was established by direct observation of divisions rather than inferred from an intestinal marker.^[ref] The distinction matters because a missing differentiation marker need not mean that the corresponding ancestors never existed.

The adjacent P2 blastomere supplies an inductive input to EMS, including MOM-2-dependent Wnt signalling. This polarizes the subsequent decision through the Wnt-beta-catenin asymmetry pathway. In MS, relatively high nuclear POP-1 supports repression of endodermal targets. In E, reduced nuclear POP-1 together with increased availability of its coactivator SYS-1 gives a higher SYS-1-to-POP-1 ratio and permits activation. Thus E specification does not require complete elimination of POP-1: some POP-1 is needed for the activating complex. Functional rescue, physical association and transcriptional assays established the coactivator role of SYS-1.^[ref]

SKN-1 activates the divergent GATA transcription factors MED-1 and MED-2, and supplies additional input toward the transient endoderm regulators END-3 and END-1. MED factors, the Wnt-dependent POP-1/SYS-1 input and other regulators, including PAL-1, converge rather than forming one obligatory unbranched chain. END-3 helps activate END-1, and the two have overlapping specification functions. Their output includes ELT-7 and ELT-2, which stabilize and execute an intestinal differentiation programme through feedback and activation of differentiated targets. Perturbing MED-binding sites in the end genes produces graded, variable gut defects, demonstrating why a binary gut/no-gut score can conceal network robustness.^[ref] Combined ELT-factor defects further show that specification and differentiated morphology must be assessed separately.^[ref]

First establish what the mutant actually loses. I would backcross the temperature-sensitive allele, measure penetrance at permissive and restrictive temperatures, and compare mutant embryos with wild-type siblings exposed to the same temperatures. Differential-interference-contrast imaging combined with a nuclear lineage label would follow EMS, its division orientation, E-cell birth, subsequent divisions, internalization and survival. I would score cell death and cell-cycle delay, not simply missing gut granules. Independent readouts would include endogenous transcripts encoding END-1 and END-3, ELT-2/ELT-7 expression, an intestinal structural marker and later intestinal morphology or lumen formation. Endogenous transcript detection controls for a transgene being selectively silenced or delayed. Stage matching by division history prevents temperature-dependent developmental rate from masquerading as altered expression timing. Loss of E-specific expression with ectopic MS-like descendants would support a fate transformation; normal early E identity with later degeneration or failed lumen formation would support a maintenance or differentiation defect.

Reciprocal temperature-shift experiments locate a functional interval. embryos maintained permissively would be shifted up at successive lineage-defined stages; a second series would start restrictively and be shifted down. Short restrictive pulses would refine any interval around the EMS division, early E divisions or later differentiation. An early upshift that causes gutlessness, a later upshift that does not, and rescue by a sufficiently early downshift would identify a temperature-sensitive period. If late shifts still damage an already marked intestine, an additional maintenance role is likely. I would measure recovery of a functional tagged GUT-1 protein, if available, because inactivation, refolding and protein perdurance broaden the observed interval. These experiments estimate when function is required, not necessarily when the gene is first transcribed. Wild-type pulse controls, matched handling and multiple clutches distinguish conditional loss of function from heat damage. Reciprocal crosses and separate shifts of mothers during oogenesis versus embryos would test maternal contribution and distinguish maternal-effect from zygotic requirements.

The site of expression and the site of requirement need separate tests. A rescuing endogenous knock-in could reveal whether GUT-1 is present in P2, EMS/E or surrounding tissue, but expression is not proof of cell autonomy. I would test rescue restricted to the signalling lineage versus EMS/E using promoters active before the inferred requirement; each construct must demonstrably produce functional protein at the relevant time. Late intestinal promoters cannot exclude an early E-lineage requirement if they fail to rescue. Marked genetic mosaics provide an independent test, provided loss of the rescuing element occurs before the sensitive interval and residual maternal protein is considered. With a suitable conditional system, local inactivation in P2 versus EMS would complement these experiments. Systemic RNA interference alone would not establish which cell requires the gene.

The most direct induction experiment would isolate EMS and recombine it with P2 in four genotype combinations. With otherwise viable blastomeres, mutant P2 failing to induce wild-type EMS suggests a signalling defect; wild-type P2 failing to induce mutant EMS suggests a responding-cell or downstream defect. Wild-type P2 plus wild-type EMS is the positive control; isolated EMS and EMS recombined with an irrelevant blastomere are negative controls. Contacts must be established at equivalent stages and orientations, with cleavage and survival recorded. The classic observation that P2, but not the other early neighbours, restores gut differentiation to isolated EMS establishes the logic of this assay.^[ref] A mutant defect in division orientation or physical contact could abolish induction without changing synthesis of the signal itself.

Finally, I would combine temporal readouts with epistasis and controlled genetic bypass experiments. Compare POP-1/SYS-1 asymmetry, MED activity, END-1/END-3 activation and ELT output in gut-1 embryos. Then express an END factor transiently in the correct early lineage, or an ELT factor at an appropriate later stage, while keeping endogenous controls and matched expression levels. Rescue by END-1 of missing end-gene activation would place the defective function upstream of, or parallel to an input bypassed by, END-1. Normal end-gene activation with failure of ELT induction would point later; persistence of early regulators with loss only of structural targets would point later still. Lack of rescue is interpretable only if the transgene is functional and the cells retain developmental competence. Because POP-1 can both repress and activate, simple POP-1 removal can produce misleading suppression or enhancement; its molecular output must be measured. Double-mutant enhancement could reveal parallel inputs rather than a linear order. Mapping, sequence identification, independent alleles and rescue by a wild-type genomic copy would establish the responsible locus. Direct promoter occupancy, a rapid transcriptional response and endogenous binding-site mutation would be needed before claiming direct regulation of a known target. The outcome should be a tested temporal and cellular requirement with constrained network relationships, not a position assigned from a single rescue.

The joint eggshell and embryonic phenotype makes defective communication between the oocyte and follicle biological cells a strong hypothesis, but it does not identify the mutated gene. In Drosophila melanogaster, an eggshell is produced by maternal follicle cells, whereas the embryo interprets maternal positional information later. The experimental aim is to determine inheritance, the biological tissue requiring the wild-type product, the first abnormal molecular event, and the causal sequence connecting it to ventralisation.

First establish a reproducible phenotype. Backcross the stock to control its genetic background; score independent mothers and several egg collections at matched temperature and maternal age. Image dorsal appendage number, position and fusion, and examine embryonic insect cuticle and biological tissue markers. Record unhatched and fertilised eggs separately. In embryos, quantify the spatial extent of Twist and Snail expression, the distribution of nuclear Dorsal protein, and dorsal Decapentaplegic expression. Expanded ventral molecular identities, rather than lethality or abnormal shape alone, establish embryonic ventralisation. Appendage defects alone are insufficient because their formation has additional regulatory requirements.

Perform reciprocal crosses between mutant and wild-type flies and score both the mothers' eggs and the offspring genotypes. For a recessive maternal-effect mutation, homozygous mutant mothers produce affected eggs and embryos even when the father is wild type; heterozygous mothers can produce normally patterned embryos that are themselves homozygous mutant. A zygotic-effect mutation instead tracks the embryo's genotype. Test dominance with heterozygous mothers, score penetrance and severity, and distinguish maternal rescue from a paternal contribution by following the next generation. Do not impose a simple Mendelian ratio if the lesion is sex-linked, semidominant or affects fertility.

Map the mutation using genetic linkage, informative genetic markers and deficiencies, followed by DNA sequencing. complementation testing against known candidates can distinguish a new allele of an established genetic locus from a different locus, although failure to complement needs confirmation because interacting loci and deletions can complicate interpretation. Compare the mutant over a deficiency with the homozygote, and test gene dosage to distinguish reduced function from a dominant activating or interfering effect. The decisive causal package is rescue by a suitably expressed wild-type genomic copy, recreation of the candidate sequence change in a clean background, and reversal by precise repair. Check that rescue restores both eggshell and embryonic pattern, rather than merely viability.

The normal information flow supplies a series of discriminating readouts. Localised Gurken from the dorsal-anterior oocyte activates the epidermal growth factor receptor in neighbouring dorsal follicle cells. This represses Pipe there, leaving its activity ventrally restricted.^[ref] The modified extracellular environment biases a proteolytic cascade that activates Spätzle; this ligand activates Toll receptor ventrally. Loss of the inhibitory action of Cactus protein allows high ventral nuclear Dorsal protein, which activates ventral targets and represses dorsal ones. Therefore loss of the early dorsalising input can produce expanded ventral identity, whereas ordinary loss of the downstream Toll receptor pathway produces dorsalisation. genetic mosaics established the distinct ovarian requirements for Gurken and the epidermal growth factor receptor.^[ref]

Use genetic mosaics to make mutant germ biological cells within wild-type ovarian soma, and mutant follicle cell clones around wild-type germ biological cells. Mark clones independently and verify their size and location. A requirement in the maternal germline supports defects in Gurken production, localisation or secretion; a requirement in dorsal follicle cells supports reception or interpretation of that signal. Include wild-type clones generated by the same procedure and tissue-specific wild-type rescue. If both compartments contribute, test whether the locus has two functions rather than forcing it into one category.

Then localise the first defect during oogenesis. Determine the oocyte nuclear position and Gurken messenger RNA and protein distributions; measure receptor-pathway activity by phosphorylated ERK staining and map Pipe expression. Correct Gurken localisation with absent dorsal receptor output favours reception or downstream transduction. Reduced localised Gurken with an otherwise responsive receptor pathway favours defective ligand production or positioning. Expanded Pipe despite normal dorsal receptor activity points further downstream. Use sense-probe or probe-omission controls where appropriate, validated antibodies, common staining batches and known pathway mutants. Temporal staging matters: a late normal-looking pattern cannot exclude an earlier transient failure.

Apply epistasis and bypass tests with explicit predictions. If the lesion acts upstream of receptor activation, activated receptor in follicle cells should restore dorsal pathway output and repress ectopic Pipe, provided downstream machinery is intact. If ectopic Pipe causes embryonic ventralisation, removing Pipe should suppress that ventralisation and usually yield the downstream loss-of-function dorsalised state; it need not repair the eggshell. Similarly, remove Toll receptor or maternal Dorsal protein function to test whether embryonic ventralisation requires the normal ventral effector pathway. Persistence after a verified downstream block demands an alternative mechanism or multiple lesions. Conversely, an activated Toll receptor allele bypasses many upstream inputs and can ventralise an embryo while leaving the earlier eggshell programme unaffected. Suppression of an embryonic defect and restoration of the eggshell are different rescue criteria.

Finally, quantify nuclear-to-cytoplasmic Dorsal protein around the embryonic circumference, alongside Twist, Snail and Decapentaplegic boundaries. Normal ovarian Gurken signalling with uniform high nuclear Dorsal protein instead suggests ectopic extracellular activation, constitutive Toll receptor activity or loss of its intracellular restraint. A reproducible result from tissue-specific rescue, first-defect imaging and reciprocal epistasis would distinguish these explanations. Neither resemblance to a known mutant nor a candidate sequence variant alone establishes how this novel lesion acts.

Let $I,J$ denote the two ordered alleles at the genetic locus. Under Hardy-Weinberg equilibrium they are independent with probabilities ${\pi }_i,{\pi }_j$. Write the multiplicative penetrance as $\mathbb{P}(D\mid I=i,J=j)=k{\psi }_i{\psi }_j$, with parameters chosen so these are valid probabilities, and set $Z={\sum }_u{\pi }_u{\psi }_u$. ThenThe Bayes' theorem givesThus multiplicative penetrance preserves Hardy-Weinberg equilibrium: the affected subjects still have two independent allele draws, with tilted allele frequencies. The displayed product uses ordered allele slots. For the usual unordered genotypes, the corresponding probabilities areThe factor of two is required for heterozygotes and must not be dropped when translating the ordered notation into genotype counts.

For genetic association analysis, the case genotype probabilities factor into allele frequencies, so under this model an allelic comparison with representative population controls is appropriate; separate dominance departures are not required by the penetrance model. The ratio of case allele frequency to population allele frequency is proportional to ${\psi }_i$, and relative risk multipliers satisfy ${\psi }_i/{\psi }_j=({\pi }_i^{\ast }/{\pi }_i)/({\pi }_j^{\ast }/{\pi }_j)$. Population controls are important to this exact statement. Among specifically unaffected controls,which generally does not factor. For a rare disease it is close to the population genotype law, but no rare-disease assumption was needed for the case factorization itself. Population stratification and sampling dependence must also be addressed before using an ordinary allelic chi-squared test.

Label each parent's two allele copies by its realized transmission: let $(A,B)$ be the transmitted and untransmitted alleles of the first parent and $(C,D^{\prime })$ those of the second. Under random mating and Hardy-Weinberg equilibrium, the four parental allele draws are independent with population frequencies $\pi$. Random Mendelian segregation merely swaps the two independent copies within each parent, so the transmitted/untransmitted labelled pair still has probability ${\pi }_a{\pi }_b$; the second pair has probability ${\pi }_c{\pi }_d$.

The child receives $A,C$. If $\mathcal{D}$ is its disease event, its penetrance is $k{\psi }_A{\psi }_C$ and its overall disease probability is $k{Z}^2$. ThereforeThis proves the transmitted and untransmitted alleles under multiplicative penetrance factorization. Under these assumptions, the transmitted pair $a/c$ has the affected-case distribution, while the complementary pair $b/d$ has the population distribution and is independent of the transmitted pair. It is the family pseudo-control genotype.

As in part (a), the formula records labelled transmissions. When observed genotypes are unordered, sum over every compatible parental-origin transmission. This includes multiplicities for overlapping parental alleles or two heterozygous parents producing a heterozygous child; it avoids interpreting one product as the total probability of all observationally identical configurations. The homogeneous random-mating model is also essential to the unconditional independence: mixing ancestry strata can correlate the case and pseudo-control through their shared family background.

The usual transmission disequilibrium test conditions on the observed parental genotypes, rather than modelling their population probabilities. For a heterozygous parent carrying $i,j$, the two transmission outcomes are equally likely under no association. With multiplicative penetrance, ascertainment through an affected child weights them by ${\psi }_i,{\psi }_j$, soThe other parent's penetrance factor cancels. This is a special case of disease-ascertained transmission probability. Under the null ${\psi }_i={\psi }_j$, the conditional probability is $1/2$. Count the discordant transmitted/untransmitted pairs from heterozygous parents and test their balance, using McNemar's test or its exact conditional binomial form. Homozygous parental transmissions do not distinguish the alternatives.

This conditional analysis removes the nuisance population allele frequencies and retains the family matching. It is therefore robust to population stratification, which can otherwise make allele frequencies differ between cases and controls without a within-family transmission effect. It does not require the unconditional Hardy-Weinberg equilibrium and random-mating assumptions used for the population pseudo-control factorization. Independence of suitably sampled families, valid genotypes and the Mendelian null transmission model still matter.

Let $G$ count copies of allele $2$. Under Hardy-Weinberg equilibrium, $G$ has a binomial distribution with parameters $2,\varphi$, and the penetrance is $q_G=p{\theta }^G$. ConsequentlyThus the overall penetrance isThe multiplicative diallelic penetrance model requires $0\le \varphi \le 1$, $p\ge 0$, $\theta \ge 0$, and each occupied genotype's value $p{\theta }^G$ to be at most one. These are probabilities, not unrestricted relative-risk scores.

The nine maximal cliques of this triangulation areEvery other nonempty clique is a subset of one of these. One junction tree, listing its separators after the edges, isEach variable appears in a connected subtree, verifying the running intersection property. For example, the cliques containing $M$ are $K_1,K_2,K_4,K_5,K_6$, and their connecting paths contain $M$ throughout.

The junction tree algorithm assigns every founder factor and every child-with-parents factor to one containing clique, and multiplies any observation or penetrance factors there. The junction-tree sum-product message from clique $i$ to clique $j$ sums out $C_i\setminus (C_i\cap C_j)$ after multiplying the local potential and all incoming messages except $j$'s. An inward pass sums the joint probability or pedigree likelihood; a subsequent outward pass gives each person's posterior genotype probabilities. This is the same variable-elimination principle as the Elston-Stewart algorithm, organized to handle the pedigree loop. Its largest clique has four genotype variables, much smaller than summing naively over fourteen at once.

The disease-ascertained transmission probability follows directly from Bayes' theorem. Assume that, given the child's genotype $g$, its disease probability is a common baseline penetrance times the genotype relative risk $R_g$, with no additional dependence on the parental genotypes. Write $q_g=P(g\mid g_f,g_m)$ for the Mendelian segregation probability. ThenThere are four equally likely labelled parental transmission outcomes. For one specified outcome $t$, this becomes $P(t\mid g_f,g_m,D)=R_{g(t)}/{\sum }_{t^{\prime }}{R}_{g(t^{\prime })}$, which is the four-outcome interpretation of the displayed formula.

For an unordered child genotype, let $m_g$ be the number of those outcomes producing it. Its probability is insteadFor example, two heterozygous parents have multiplicities $(1,2,1)$ for $11,12,22$. A heterozygous child therefore has probability $2R_{12}/(R_{11}+2R_{12}+R_{22})$. The four possible outcomes must be counted with multiplicity; treating their unordered genotypes as four distinct equally likely values would be incorrect.

For the Mendelian identity by descent count $J$, the expected shared proportion isThe corresponding expected number of shared copies is one. These are different scales for the same random variable.

With genetic linkage to a disease locus, selecting two affected full siblings preferentially selects transmissions of disease-associated ancestral copies. A nearby genetic marker may then show excess identity by descent: the expected shared proportion can exceed $1/2$, which is the basis of affected-sib-pair analysis. The precise distortion depends on penetrance, inheritance and the recombination fraction; disease status alone does not prescribe a universal alternative distribution. In the fully penetrant recessive model derived below, setting $s=2\theta (1-\theta )$ givesThus the expected shared proportion rises from $1/2$ at $\theta =1/2$ to one at complete genetic linkage, $\theta =0$.

Parametric linkage analysis specifies a disease-transmission model: the mode of inheritance, disease-allele frequency and penetrance of each genotype. Combining these with Mendelian segregation and the recombination fraction gives a pedigree likelihood. One can estimate the recombination fraction or compare a linked model, $\theta <1/2$, with the unlinked model, $\theta =1/2$, using a likelihood-ratio test or LOD score. Incomplete penetrance and unobserved genotypes are accommodated by summing over the possible underlying inheritance states.

Nonparametric linkage analysis instead tests whether relatives with the same disease phenotype share marker alleles identical by descent more often than expected from their pedigree. For example, affected-sib-pair analysis compares sharing among affected full siblings with its Mendelian null distribution. It avoids specifying a complete disease-penetrance model and disease-allele frequency. “Model-free” still assumes an inheritance model for the marker, correct family relationships and a justified treatment of genetic-study ascertainment; it does not mean assumption-free.