1.1. Mathematical Approaches to Predictive Biological Science
1.2. Entropy, Information, and Related Approaches in Genetics
- incorporability into a model-fitting or statistical testing framework.
2. Ability to Express Diversity in a Way That Makes Intuitive Sense
2.1. Diversity Measures and Partitioning Diversity in Genetics and Ecology
- Variety–“the number of categories into which system elements can be apportioned” . This is also called “richness” in biology, e.g., the number of different allelic types or the number of different species, termed S in this article.
- Balance–“a function of the pattern of apportionment of elements across categories” . This is based on pi–e.g., the proportions of each different type of allele.
- Disparity–“the manner and degree in which the elements may be distinguished” . This has been given a large number of names in biology, some of which will be introduced later.
- DNA sequence diversity and linkages along DNA in the genome;
- Sequence diversity between different alleles within one individual, for organisms with more than one genome (e.g., diploids such as humans);
- Diversity of alleles within one population;
- Diversity of allele proportions in different populations of the same species;
- Diversity of interactions between genes and environmental factors;
- Diversity of genetics, morphology, etc., between species;
- Diversity of types of species within a community;
2.2. “One-part” Diversity Measures
|Allele Xa||Allele Xb|
|1||fa1 = N1pa1||fb1 = N1pb1||N1 = r1 Ntot|
|2||fa2 = N2pa2||fb2 = N2pb2||N2 = r2 Ntot|
|Summed Populations, or Marginal Total||Na = N1pa1 + N2p a2||Nb = N1p b1 + N2p b2||Ntot = N1 + N2|
2.3. Two-part Approaches to Diversity
3. Integrating Genetic Diversity Measures with Natural Processes such as Selection and Dispersal
3.2. Gene Interactions
|A the SNP||GG||12||43||18|
|A the SNP||GG||12||13||18|
4. Ability to be Incorporated into an Inclusive Statistical Framework
5. Future Directions
References and Notes
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alternative versions of the DNA sequence at a locus; see pi.
“a function of the pattern of apportionment of elements across categories”. This is based on what is called in this article.
various types of selection which tend to maintain variants. Also called stabilizing selection in multilocus cases.
a component of DNA, also (somewhat loosely) called a nucleotide. There are four possible bases, A C T and G. The sequence of bases in DNA spells out the code. In the case of portions that code for amino acids, the code is read in triplets of bases.
represents some estimate of difference between the types (e.g., difference of morphology of species, or number of non-shared bases between alleles). See disparity.
This is when cells contain two genomes, one from each parent individual, so that each gene might be represented by two different alleles in the one individual. Much of the information in humans is diploid. Where there is only one genome, as for the Y-chromosome or mitochondrial or chloroplast DNA, this is called haploidy (not mono- or uni-ploidy, as one might expect!). Polyploidy is when there are more than two genomes in each cell.
|Differential entropy h|
A continuous version of Shannon entropy. See Equation 17.
carries the genetic code. Composed of bases. Some parts of the code are in triplets.
the manner and degree in which the elements of a group (e.g., the different allele types) may be distinguished. See .
selection which eliminates one genetic variant in favour of another. Also called positive selection when the focus is on the favoured variant, or negative when the focus is on the disadvantageous variant.
|Disruptive or divergent selection|
when phenotypically intermediate genotypes are at a disadvantage.
in heterozygotes, where the two different alleles from each parent are not the same type, sometimes it is only possible to detect the phenotypic effect of one allele–the dominant allele. The other allele is said to be recessive.
random processes in transmission of genes from one generation to the next.
|Effective number of alleles (entropic) neS|
neS is the number of equi-frequent alleles that would be required to provide the same value as the actual sample–see Equation 4. This is the entropic analogue of neH.
|Effective number of alleles (heterozygosity) neH|
neH is the number of equi-frequent alleles that would be needed to give the same heterozygosity as the actual sample–see Equation 2. Also see neS.
|Effective population size|
interaction between the effects of two different loci, in production of the phenotype.
a transform of one of the diversity indices (usually Shannon’s) to make explicit the departure from the most diverse case: equal numbers of each type of allele–see Equations 5 and 6.
a function of the survival and reproduction of carriers of a certain genotype. Genotypes with higher fitness will tend to become more numerous over the generations. See also “selection”.
this word is used variously to mean locus or allele. In the present review, it is restricted to meaning a protein-coding locus. The word should probably be abandoned, due to its sloppy use.
a complete set of genetic information, coded as base sequence of DNA. Some of this code is in triplets which each specify an amino acid in a protein. Other parts of the genome have other functions, such as regulating the expression of parts of the genome.
the alleles contained in an individual for one or more loci.
see Shannon entropy.
see differential entropy.
a block of DNA containing multiple SNPs, and coding for one or more genes and their regulatory regions. Haplotypes are an example of genetic linkage.
an individual whose genotype has one copy of each of two different alleles, at a diploid locus.
the chance of drawing two different alleles at random (with replacement) from a population: see Equation 1. Note that in this review, I do not also discuss the observed heterozygosity–the actual occurrence of heterozygous individuals. See supplement of Sherwin et al. 06  for more discussion of this, as well as its information- theoretic applications. Heterozygosity is also called “Simpson index” when applied to species in ecological communities, “Haplotype Diversity” when it is the chance of drawing two different haplotypes at random, or “Nucleotide Diversity” when it is the chance of drawing two different nucleotides at random.
an individual whose genotype has two copies of the same allele at a diploid locus.
an insertion or deletion which appears in one sequence when compared to another sequence. These occur naturally during evolution of DNA. During reconstruction of phylogenies, the size and relative positions of indels must be estimated in a trade-off with the number of mismatched bases at other positions .
|Infinite alleles model (IAM)|
for a given set of observed proportions of different types this is a comparison of the entropy based on an underlying distribution which really is given by versus the entropy if the underlying proportions follow some other distribution, . Also called relative entropy or information gain.
Linkage of two different genetic loci in the genome is when the inheritance of allelic variants at one locus is not statistically independent of alleles at another. Apparent linkage between two loci is called “linkage disequilibrium” or a more correct term “gametic disequilibrium” which recognizes that apparent linkage can be due to causes other than actual physical linkage.
a position in the genome. Sometimes restricted to a protein-coding region, other times applied to any region of DNA at a fixed location in the genome, such as a SNP.
a change to the genetic code. Note that this is best called a change, not an error–all current codes, advantageous and deleterious, were derived via multiple mutations. Various different types of mutation occur. Two contrasting types that are commonly modeled are infinite allele model (IAM), and stepwise mutation model (SMM). In IAM, every mutation makes a novel allele, which is a reasonable approximation of the evolution of a coding region made up of thousands of bases, each with four alternatives, A C G T, and a per-base mutation rate such as μ = 10-9 per generation. SMM or similar is seen in repetitive regions such as CACACACACACA, where repeats (CA) are added or subtracted, so that alleles of the same length are re-created regularly.
For two variables, the mutual information between them is the reduction in uncertainty of the level of one variable, when there is information about the level of the other variable. Or, roughly stated, this is the ability of one type of information to enlighten us about another. For example, if two populations have no shared genetic variants, then knowing the genotype of an individual would give a perfectly accurate guide to the individual’s population membership, so there is said to be high mutual information between the genes and the population membership. Conversely, if the two populations have exactly the same arrays of genetic variants, then knowledge of the genes gives no indication of population membership, so mutual information is zero. See Equation 9.
effective population size: This depends not only upon actual population size, but also on any other factor that alters the rate at which random processes affect genetic quantities such as the heterozygosity 
see effective number of alleles.
see directional selection.
the proportion of entities of type i in some group (e.g., numbers of different allelic variants encountered in a population, or numbers of different species encountered in an ecological community). See balance.
the detectable effect of genetic and environmental information. This might be shape, chemistry, or colour of the organism carrying a certain genotype, or the survival and reproduction of that individual.
a reconstruction of the evolutionary history of a number of separate groups, usually based only upon present-day data from those groups .
the occurrence of more than one variant within a population, e.g., two different alleles at the same locus.
see directional selection.
|Q, or Quadratic Entropy|
a generalization of Simpson’s index/Heterozygosity:
the proportion of a species that is in each of two populations 1 and 2. This may sometimes also be used as the relative sizes of the samples form the two populations, when performing significance testing. Note that these symbols are not to be confused with the correlation between uniting gametes, r2, used in linkage analysis.
a measure of the interaction between genotypes and environmental conditions, in production of phenotypes.
this occurs when two haplotypes from different genomes break and rejoin to make new combinations of the alleles at the different loci, ie new haplotypes.
a molecule similar to DNA, e.g., messenger RNA which carries the DNA code to the cell to be converted to an amino acid sequence in a protein.
see Shannon’s diversity or entropy.
the consequence of fitness differences. Genotypes with higher fitness will tend to become more numerous over the generations. See also directional, balancing and disruptive selection. Relative fitness of different genotypes is often expressed by selection coefficents s, where one genotype is arbitrarily assigned maximum fitness of 1, and other genotypes are given fitnesses reduced by a selection coefficient s, so their fitness is 1-s ().
|Shannon’s diversity or entropy|
Where there is variation at one base position, this is called a “single-nucleotide-polymorphism” or SNP. Thus the alleles of a SNP locus are alternative bases A C G or T.
after RNA code is transcribed from the DNA code, often portions of the code are removed, between two splice sites, before the code is used to direct the production of proteins.
|Stepwise mutation (SMM)|
“the number of categories into which system elements can be apportioned”. Also called “richness” in biology, e.g., the number of different allelic types or the number of different species, termed S in this article.
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