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An active site is the region of an enzyme where a substrate binds and undergoes a chemical reaction. In a protein enzyme, it includes the amino acid residues that contact the substrate and those that directly assist catalysis. These roles can overlap: an amino acid side chain can both orient a substrate and participate in chemistry. The active site occupies only part of the folded protein; the remaining amino acid residues help establish its protein structure, stability and movement. An active site need not be a consecutive stretch of protein primary structure: protein folding can bring distant amino acid residues together. A ribozyme instead supplies an RNA active site. See Alberts et al., Molecular Biology of the Cell, Protein Function: binding sites and catalytic sites.

Binding and catalysis answer different questions. A protein binding site recognizes shape and chemistry through such interactions as hydrogen bonds, electrostatics and the hydrophobic effect. At an active site, appropriate positioning also helps a substrate reach the transition state of a chemical reaction. Some amino acid side chains transfer a proton; others participate transiently in a covalent bond. The enzyme-substrate complex is the associated enzyme and substrate, before conversion to a reaction product. Merely accumulating this complex does not establish rapid catalysis: stabilizing the bound substrate without sufficiently stabilizing the transition state can leave a large barrier. The important complementarity is chemical as well as geometric. See Geoffrey M. Cooper, The Cell, The Central Role of Enzymes as Biological Catalysts: substrate binding and transition-state stabilization.

The cartoon makes a useful experimental distinction. Changing a contacting amino acid residue might weaken substrate binding, change the reaction rate, or do both. Changing a distant amino acid residue can also disrupt the active site by changing protein folding. Therefore a loss of activity after a mutation alone does not locate a catalytic group: compare binding, protein structure and enzyme kinetics as well. This is an interpretation guide, not a report of a particular mutation experiment.

A concrete example is hexokinase: it transfers the terminal phosphate group of ATP to the $6$-position hydroxyl group of a sugar such as glucose. Specificity need not mean only one acceptable substrate: the hexokinase reaction class includes several possible acceptor sugars. Which are used, and at what reaction rates, depends on the particular enzyme and conditions. See Swiss Institute of Bioinformatics, ENZYME entry EC 2.7.1.1: hexokinase reaction and acceptor range.

Bennett and Steitz compared yeast hexokinase A crystallized with glucose against hexokinase B crystallized without glucose. X-rays scattered by the crystals supplied diffraction measurements from which the authors reconstructed protein structure; comparing coordinates located movement, rather than measuring a reaction rate. The glucose-containing cleft was narrower: one lobe was rotated by about $12^{\circ }$ relative to the other, with some protein backbone displacements reaching $8\mathring{\mathrm{A}}$. This makes the active site a moving arrangement of interacting groups, rather than a permanent hole. See William S. Bennett Jr. and Thomas A. Steitz, Glucose-induced conformational change in yeast hexokinase, PNAS 75 (1978), 4848–4852.

The structural comparison has two explicit limitations: different forms of hexokinase and different crystal environments were compared, and static coordinates do not record the order of binding and movement. Closure is consistent with induced fit, but those structures alone cannot exclude conformational selection. The solution experiment under induced fit addresses whether compaction also occurs outside a crystal.

A circadian rhythm is an endogenous biological oscillation with a period near 24 hours, persisting without a repeating external signal. A circadian clock times biological activities and anticipates daily changes without continual external instructions. A free-running circadian rhythm reveals its intrinsic period; entrainment of circadian rhythm coordinates it with an external cycle.^[ref]

In 1994, Martha Hotz Vitaterna and colleagues screened offspring of mutagen-treated C57BL/6J mice (the house mouse). Each mouse lived in a cage with a running wheel, inside a ventilated, light-tight chamber with timer-controlled lighting. Wheel revolutions were recorded electronically or on an event recorder. The initial screen used 14:10 LD for the first 41 animals and 12:12 LD thereafter; LD denotes hours of light and darkness. Transfer to constant darkness, DD, omitted the next scheduled lights-on; recording continued for at least three weeks. The first 41 animals also received brief light pulses, so those records were not uninterrupted DD.^[ref]

The subsequent-generation comparison in Figure 2 used 12:12 LD before DD. Animals without the Clock mutation and with mutant genotypes supplied the comparison: the B6 examples in panels A–C had initial DD periods of 23.6, 24.8 and 27.1 hours, respectively; the homozygote later lost its sustained circadian rhythm. Total activity remained comparable across genotypes in the interstrain comparison. The mutation therefore altered timing, beyond merely reducing running. Wheel activity is a behavioral readout; it does not directly measure every circadian clock or establish a human period.^[ref]

Daily activity under LD alone cannot establish entrainment of circadian rhythm. Light can directly change activity without resetting the circadian clock: masking of circadian rhythm. Removing the light cycle helps distinguish these effects; they can coexist.^[ref] In an experiment, other repeating inputs must also be controlled: a nominally dark room with daily handling is not automatically an environment without timing cues.

A mutation affecting a house mouse circadian clock.

Clonal development builds a group through successive divisions of founding biological cells, rather than by gathering previously separate lineages. In snowflake yeast, daughter biological cells remain attached after budding. Common ancestry does not imply that every descendant remains genetically identical forever: new mutations can arise. See the 2015 study cited under multicellular organism.

Effective population size is the size of an idealized population giving the relevant rate of genetic drift or ancestral coalescence in the population being modeled. In the standard neutral diploid coalescent, time is measured in $2N_e$ generations and the population-scaled mutation parameter is $\theta =4N_e\mu$.

// Target: biology.bigb

Evolution is descent with inherited modification. Within a population, changes in allele frequencies provide a precise genetic description; over longer histories, branching descent connects populations through common descent. An individual growing larger is biological development, whereas an inherited variant becoming more frequent across generations is evolution. See UC Museum of Paleontology: the definition of evolution.

Mutation supplies new alleles, genetic recombination reshuffles inherited material, natural selection favors variants with greater reproductive contribution in a specified environment, and genetic drift changes frequencies through chance. Gene flow moves variants between populations. Consequently, an increasing allele frequency alone does not identify natural selection. See Clark, Douglas and Choi, Biology 2e: population genetics.

Evidence includes ordered fossils, shared inherited structures and similarities in DNA; phylogenetics uses such observations to infer relationships. A phylogenetic tree is an inferred historical account, with uncertainty about particular branches. See Clark, Douglas and Choi, Biology 2e: evidence of evolution. Evidence also comes from experiments: Blount, Borland and Lenski reported an Escherichia coli lineage able to use citrate with oxygen present by 31,500 generations, and tested historical dependence by replaying evolution from stored ancestors. See Blount, Borland and Lenski, 2008: original experimental report. Such observations constrain explanations; no single frequency equation captures every evolutionary process.

Experimental tractability is the practical ability to cultivate a system, manipulate the variables of interest, and obtain an interpretable readout with appropriate scientific controls. In Saccharomyces cerevisiae, inexpensive cultivation, accessible haploid and diploid states, and tools for introducing DNA make controlled changes of genotype feasible. A haploid biological strain exposes a changed allele without a second copy at that genetic locus, while crosses and meiosis permit tests of inheritance. ^[ref]

An accessible readout is not necessarily a direct measurement of the mechanism. Growth of a biological colony integrates many cellular functions. Failure to grow could reflect the intended gene deletion, unsuitable conditions, failed genetic transformation, or another mutation. Matching the genetic background and using scientific controls narrows these alternatives. Ease of doing an experiment and strength of its inference are distinct advantages.

Experimental Reconstruction of Functional Gene Transfer from the Tobacco Plastid Genome to the Nucleus by Sandra Stegemann and Ralph Bock (2006, Results, pp. 2870–2874) separated relocation of DNA from activation of a relocated gene. Engineered tobacco chloroplasts carried an nptII gene with nuclear expression signals beside an aadA gene with plastid expression signals. Selection on kanamycin recovered plants with the segment in the nuclear genome. Crosses then replaced the engineered chloroplasts with ordinary chloroplasts: the nuclear segment remained, but the plants were initially sensitive to spectinomycin and streptomycin. The transferred aadA gene was not yet functional in its new setting.

Leaf pieces were cultured on regeneration medium containing both drugs. Resistant regenerated plants supplied the functional readout. Using both drugs reduced selection of spontaneous plastid mutations conferring resistance mainly to one drug. Reciprocal crosses and selfing distinguished maternal plastid inheritance from nuclear Mendelian segregation; eight of sixteen initial candidates had the latter pattern. ^[ref]

The molecular readout came from RNA hybridization and DNA sequencing. A Northern blot detected aadA messenger RNA in activated lines, with different transcript sizes. Thermal asymmetric interlaced polymerase chain reaction recovered upstream DNA; products present in activated lines but absent in original-line and ordinary-plant controls were sequenced. The mapped rearrangements connected aadA to the upstream nuclear gene promoter. Thus resistance, inheritance, RNA detection and changed nucleotide sequences supported activation of the transferred nuclear gene, rather than resistance alone. ^[ref]

This engineered test demonstrates one route to functional nuclear gene expression after plastid DNA transfer. It does not establish a natural transfer rate, replay ancient endosymbiosis or demonstrate that the resulting protein is imported into a chloroplast. The authors explicitly identify acquisition of targeting information as an additional step not reproduced by this experiment. ^[ref]

Genetic recombination rearranges inherited genetic material into new combinations. In the meiosis example considered here, chromosomal crossover exchanges segments between nonsister chromatids of homologous chromosomes. A transmitted chromatid can consequently contain DNA from both parental homologues. This produces new haplotypes without requiring a new mutation at either scored genetic locus. See Meiosis.

For phase-known genotype $AB/ab$, a single chromosomal crossover between the selected genetic loci can yield $Ab$ and $aB$ on the participating chromatids; the other two chromatids can remain $AB$ and $ab$. Thus an exchange in the four-chromatid bundle does not mean every resulting gamete is recombinant. Only a chromosomal crossover encountered along the transmitted chromatid within the selected interval affects its two-locus score.

Follow parental origin along the transmitted chromatid from the first genetic locus to the second. Each exchange switches that origin. An odd number of such chromosomal crossovers changes the endpoint combination; an even number restores parental phase. Two exchanges can therefore leave the scored haplotype as $AB$ even though an internal segment came from the other parental homologous chromosome. Recombinant status is relative to the chosen genetic loci, not a complete record of exchanges along the chromosome.

This integer parity distinction connects the physical process of genetic recombination to the observed recombination fraction. It also explains why a long genetic map interval can accumulate many exchanges while its endpoint recombination fraction approaches $1/2$ in the Haldane mapping function model.

Consider an explicitly idealized haploid population with two inherited types $A,B$, one allele per individual at the tracked genetic locus, and discrete, nonoverlapping generations. Assume faithful type transmission, no mutation or gene flow, a constant environment, and reproductive contributions independent of type frequency. Let the expected values of offspring surviving to the next census be $m_A,m_B>0$, with relative genotype fitness $w=m_A/m_B$.

For $A$ frequency $p$, expected offspring contributions are proportional to $p{m}_A$ and $(1-p)m_B$. Normalizing gives the deterministic large-population mathematical modelThe denominator is mean relative genotype fitness. For $w=1.2$ and $p_0=0.2$, $p_1=0.24/1.04=3/13\approx 0.23077$. Equivalently, 200 $A$ and 800 $B$ parents with $m_A=1.2,m_B=1$ contribute expected counts 240 and 800. Fractional per-parent expected values are averages, not literal fractional offspring. The ratio of these expectations is a deterministic approximation, not generally the expectation of a random ratio.

The odds multiply particularly simply:Thus $p_{10}\approx 0.60753$: repeated modest advantage can substantially change composition. For $w>1$ and $0<p<1$, $p^{\prime }-p=(w-1)p(1-p)/(1+(w-1)p)>0$. The deterministic boundary $p=0$ remains unchanged: natural selection needs existing variation.

A distinct stochastic extension fixes $N$ offspring and samples their types independently using a binomial distribution: $K\sim \mathrm{Binomial}(N,q)$, where $q=wp/(wp+1-p)$, then $p^{\prime }=K/N$. Its conditional expected value is $q$ and conditional variance is $q(1-q)/N$. With $w=1$, the conditional expected value remains $p$, yet realized frequencies wander through genetic drift. With $w>1$, genetic drift can still lower $p$ or lose $A$ entirely. Over several generations, the deterministic curve need not equal the mean stochastic trajectory, because the recurrence is nonlinear.

The Hardy-Weinberg principle describes the genotype proportions produced by independently combining two parental gametes from the same allele pool. For alleles $A,a$ with allele frequencies $p,q$, where $p+q=1$, the Hardy-Weinberg proportions areThe two ordered heterozygote outcomes contribute $pq+qp$. These proportions sum to $(p+q{)}^2=1$, and their implied $A$ allele frequency is $p^2+(2pq)/2=p$. Reshuffling copies therefore preserves their frequencies.

For stable Hardy-Weinberg equilibrium across generations, use a large population with random mating, equal allele frequencies in the parental pools, ordinary Mendelian segregation, and no natural selection, mutation, gene flow or genetic drift. Negligible genetic drift is an idealization of large population size. The one-generation random-mating equilibrium needs independent union of gametes; persistence additionally requires processes that change allele frequencies to be absent.

At $p=0.70,q=0.30$, the predicted genotype proportions are $0.49,0.42,0.09$. For 100 sampled individuals the expected values of the counts are $49,42,9$, rather than the $25,50,25$ of a specified $Aa\times Aa$ cross. The latter fixes both parental genotypes; Hardy-Weinberg proportions average over population mating.

An observed population need not have these proportions. For example, $70AA$, no $Aa$, and $30aa$ among 100 individuals still gives $p=0.70$, but has no heterozygotes. Random mating predicts $0.42$ heterozygosity in the next generation under the stated model. A finite sample can also depart by chance, so a discrepancy is a starting point for investigating inbreeding, population stratification or genotyping error, rather than proof of one cause.

Hardy's 1908 note made the conceptual payoff explicit: a dominant phenotype does not by itself make its allele spread. Complete dominance describes expression; changes in frequency require a separate mechanism such as natural selection. See Hardy, Mendelian Proportions in a Mixed Population, Science 28 (1908), 49–50.

The epoch lengths in Kingman's coalescent are independent random variables. Conditional on these lengths, mutation counts on disjoint tree pieces are independent Poisson random variables. Their conditional laws depend only on the corresponding epoch length, so integrating the product of conditional laws against the independent epoch laws leaves independent unconditional epoch mutation counts. This justifies adding their variances, rather than assuming independence from conditional independence alone.

Genetic recombination erodes linkage disequilibrium by reshuffling alleles between haplotypes, while preserving marginal allele frequencies in the idealized model. Let $f_{AB,t}$ denote the haplotype frequency in the gamete pool of generation $t$, let $p,q$ denote its marginal allele frequencies, and let $r\in [0,1/2]$ be the constant recombination fraction between two autosomal genetic loci. Assume random mating within one population, nonoverlapping generations, fair Mendelian segregation, equal reciprocal gamete production, the same $r$ in both sexes, and a deterministic population large enough to neglect genetic drift. Exclude natural selection, mutation and gene flow. ThenHere $p$ and $q$ remain constant: recombining changes which alleles occur together, not their total numbers. The other haplotype frequencies change by $+r{D}_t$ for $Ab,aB$ and $-r{D}_t$ for $ab$. Consequently the joint probability distribution stays within its allowed bounds and approaches the product of its marginal distributions when $r>0$.

For the original positive example $p=0.60,q=0.40,D_0=0.12$, a recombination fraction $r=0.10$ gives $D_1=0.108$ and $D_{10}\approx 0.04184$. The negative example $D_0=-0.12$ has exactly the opposite sequence: genetic recombination reduces the magnitude without reversing the sign. With $r=0$, the initial linkage disequilibrium persists in this model; even independent assortment at $r=1/2$ only halves $D$ each generation. A population association does not vanish after one generation merely because the genetic loci are unlinked.

The geometric sequence gives a generation scale: for $r>0$, the continuous interpolation reaches $|D_0|/2$ at $t_{1/2}=\log (1/2)/\log (1-r)$. Actual generations are integers; the first at or below half is $\lceil t_{1/2}\rceil$. For small $r$, $t_{1/2}\approx (\log 2)/r$. Because the denominator of allelic correlation stays fixed, $r_{\mathrm{LD},t}^2=(1-r{)}^{2t}{r}_{\mathrm{LD},0}^2$; its decay is faster than the magnitude of $D_t$.

A short derivation uses random mating to assign frequency $2f_{AB}{f}_{ab}$ to the double heterozygote $AB/ab$ and $2f_{Ab}{f}_{aB}$ to $Ab/aB$. Relative to $r=0$, each loses or gains an $AB$ gamete fraction $r/2$, respectively. Their net contribution is $r(f_{Ab}{f}_{aB}-f_{AB}{f}_{ab})=-rD$. The other genotypes produce no net change in $f_{AB}$, proving the recurrence and, by iteration, its solution.

This temporal model does not identify elapsed generations from physical separation alone. The recombination fraction depends on genetic recombination, including recombination hotspots, while population stratification, natural selection, gene flow or genetic drift can create or maintain linkage disequilibrium. Repeated sampling across time would also require accounting for sample size and haplotype estimation. A cross-sectional plot of association against distance between genetic loci is therefore a different observation from the idealized trajectory plotted here.

Meiosis reduces two sets of chromosomes to one. In the ordinary diploid life cycle, this makes it possible for fertilization of two haploid gametes to restore the diploid state rather than double the number of sets in every generation. Its two divisions perform different tasks: meiosis I separates homologous chromosomes, whereas meiosis II separates sister chromatids. One round of DNA replication precedes both divisions; there is no second round between them. See ^[ref].

Two homologous chromosomes carry corresponding genetic loci, potentially with different alleles; two sister chromatids originate by DNA replication of one chromosome. After chromosomal crossover between nonsister chromatids, the sister chromatids need not carry identical alleles everywhere. Separating the two relationships explains why a haploid product of meiosis I still contains replicated chromosomes.

The two divisions explain Mendelian segregation; variable orientation of different homologous chromosome pairs connects it to independent assortment. Genetic recombination adds another source of allele combinations. These mechanisms reshuffle inherited material without requiring a new mutation.

The symmetric diagram is useful for counting, but human oogenesis produces one functional ovum rather than four. The polar bodies receive discarded chromosomes and little cytoplasm. See Human Reproductive Anatomy and Gametogenesis (§43.3, Oogenesis and Figure 43.13: unequal divisions and polar bodies).

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.

Count the same hypothetical diploid animal biological cell in three ways: chromosomes, double-stranded DNA molecules, and nuclear DNA amount. Assume one continuous double-stranded DNA molecule per unreplicated chromosome or replicated chromatid, complete S phase, and no losses or mutations. Exclude mitochondrion DNA. Here $n=2$ is the size of one haploid set, and $C$ is its unreplicated nuclear DNA amount, as defined in cell-cycle DNA content. A double-stranded DNA molecule counts as one, despite containing two strands.

The allocation can be checked copy by copy. DNA replication turns $A$ into $A_1,A_2$, $a$ into $a_1,a_2$, and similarly for $B,b$. One daughter receives $A_1,a_1,B_1,b_1$ and the other $A_2,a_2,B_2,b_2$. Each therefore inherits the same four chromosome identities and the same gene and allele content as the unreplicated parent, under the stated assumptions. Across both daughters there are $4+4=8$ DNA molecules and $2C+2C=4C$: partition conserves the material present after S phase.

The transient whole-cell count of eight does not mean that each daughter's ploidy has doubled. Each completed daughter is diploid, containing both homologues of each homologous chromosome pair. This differs from separating homologous chromosomes during meiosis I. Faithful mitosis preserves a nuclear genome through growth and renewal; equal inherited genes need not imply equal gene expression or equal cell fate. Mutations or chromosome segregation errors can break the ideal inheritance pattern.

In Kingman's coalescent, the epoch with $j$ ancestral lineages lasts $T_j\sim \mathrm{Exp}((\genfrac{}{}{0ex}{}{j}{2}))$, independently across epochs. If each lineage carries an independent Poisson process of mutations with rate $\theta /2$, its epoch mutation count satisfies $Y_j\mid T_j\sim \mathrm{Poisson}(j\theta T_j/2)$. The total mutation count is $S={\sum }_{j=2}^n{Y}_j$. Under the infinite sites mutation model, this also counts segregating sites; without that assumption it still counts mutation events, which need not correspond one-to-one to observed polymorphisms.

A mutation rate measures the frequency of mutation events per specified unit, such as per chromosome segment per generation. Its units and the population-size convention must be specified. If time is measured in $N$ generations and $\theta =2Nu$, each ancestral lineage has mutation intensity $Nu=\theta /2$ per unit of coalescent time.