Independent Segregation of Chromosomes solved by 471

Dec. 4, 2012, 7:08 a.m. by Rosalind Team

Topics: Heredity, Probability

Mendel's Work Examined

Mendel's laws of heredity were initially ignored, as only 11 papers have been found that cite his paper between its publication in 1865 and 1900. One reason for Mendel's lack of popularity is that information did not move quite so readily as in the modern age; perhaps another reason is that as a friar in an Austrian abbey, Mendel was already isolated from Europe's university community.

It is fair to say that no one who did initially read Mendel's work fully believed that traits for more complex organisms, like humans, could be broken down into discrete units of heredity (i.e., Mendel's factors). This skepticism was well-founded in empirical studies of inheritance, which indicated a far more complex picture of heredity than Mendel's theory dictated. The friar himself admitted that representing every trait with a single factor was overly simplistic, and so he proposed that some traits are polymorphic, or encoded by multiple different factors.

Yet any hereditary model would ultimately be lacking without an understanding of how traits are physically passed from organisms to their offspring. This physical mechanism was facilitated by Walther Flemming's 1879 discovery of chromosomes in salamander eggs during cell division, followed by Theodor Boveri's observation that sea urchin embryos with chromatin removed failed to develop correctly (implying that traits must somehow be encoded on chromosomes). By the turn of the 20th century, Mendel's work had been rediscovered by Hugo de Vries and Carl Correns, but it was still unclear how Mendel's hereditary model could be tied to chromosomes.

Fortunately, Walter Sutton demonstrated that grasshopper chromosomes occur in matched pairs called homologous chromosomes, or homologs. We now know that the DNA found on homologous chromosomes is identical except for minor variations attributable to SNPs and small rearrangements, which are typically insertions and deletions. Sutton himself, working five decades before Watson & Crick and possessing no real understanding of DNA, actually surmised that variations to homologous chromosomes should somehow correspond to Mendel's alleles.

Yet it still remained to show how chromosomes themselves are inherited. Most multicellular organisms are diploid, meaning that their cells possess two sets of chromosomes; humans are included among diploid organisms, having 23 homologous chromosome pairs.

Gametes (i.e., sex cells) in diploid organisms form an exception and are haploid, meaning that they only possess one chromosome from each pair of homologs. During the fusion of two gametes of opposite sex, a diploid embryo is formed by simply uniting the two gametes' halved chromosome sets.

Mendel's first law can now be explained by the fact that during the meiosis each gamete randomly selects one of the two available alleles of the particular gene.

Mendel's second law follows from the fact that gametes select nonhomologous chromosomes independently of each other; however, this law will hold only for factors encoded on nonhomologous chromosomes, which leaves open the inheritance of factors encoded on homologous chromosomes.

Problem

Consider a collection of coin flips. One of the most natural questions we can ask is if we flip a coin 92 times, what is the probability of obtaining 51 "heads", vs. 27 "heads", vs. 92 "heads"?

Each coin flip can be modeled by a uniform random variable in which each of the two outcomes ("heads" and "tails") has probability equal to 1/2. We may assume that these random variables are independent (see “Independent Alleles”); in layman's terms, the outcomes of the two coin flips do not influence each other.

A binomial random variable $X$ takes a value of $k$ if $n$ consecutive "coin flips" result in $k$ total "heads" and $n-k$ total "tails." We write that $X \in \mathrm{Bin}(n, 1/2)$.

Given: A positive integer $n \leq 50$.

Return: An array $A$ of length $2n$ in which $A[k]$ represents the common logarithm of the probability that two diploid siblings share at least $k$ of their $2n$ chromosomes (we do not consider recombination for now).

Sample Dataset

5

Sample Output

0.000 -0.004 -0.024 -0.082 -0.206 -0.424 -0.765 -1.262 -1.969 -3.010

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