1Foundation of Mathematical Biology / Foundation of Mathematical Biology
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How good is the test? |
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In large normal samples, the t test is slightly better at finding significant differences
In small non-normal samples, the rank sum test is rarely much worse than the t test and is often much better
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Comparing distributions |
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Suppose we want to know if there is any difference between the distributions of two sets of observations
We don’t care if the difference is location or dispersion The Kolmogorov-Smirnov test
♦Informally: related to the maximum difference between the cumulative histograms of the two sample sets
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chist( pop1 ) − chist( pop2 ) |
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gcd(m, n) |
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Again, look up whether J is big enough to reject the null hypothesis that the distributions are the same.
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Informal example: Relationship of genomic copy number to gene expression
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Example: Kolmogorov-Smirnov test |
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We are looking at the ability of people to generate saliva on demand, plus and minus feedback to tell them if they are successful.
Our max chist difference is 6/10.
Our multiplier (mn/(gcd(m,n)) is (10*10/10 = 10)
So J = 6. From a table, we get p = 0.0524
We sort all of our samples.
We compute the cumulative histogram using the values from each set as the thresholds (since these are the only points where a change will happen).
We find the max difference.
UCSF
Molecular similarity: Quantitative comparison of 2D versus 3D
Nicotine example
♦Nicotine
♦Abbott molecule: competitive agonist
♦Natural ligand (acetylcholine)
♦Pyridine derivatives
2D similarity
♦Graph-based approach to comparing organic structures
♦Very efficient algorithm
♦Can search 100,000 compounds in seconds
Ranked list versus nicotine places competitive ligands last
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1.00 |
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0.99 |
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0.82 |
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0.73 |
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0.57 |
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0.54 |
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00..1313 |
UCSF |
Molecular similarity: 2D versus 3D |
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Nicotine example
♦Nicotine
♦Abbott molecule: competitive agonist
♦Natural ligand (acetylcholine)
♦Pyridine derivatives
3D similarity
♦Surface-based comparison approach
♦Requires dealing with molecular flexibility and alignment
♦Much slower, but fast enough for practical use
Ranked list places the Abbot ligand near the top, and acetylcholine has a “high” score
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1.00 |
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0.90 |
0.89 |
0.880.88 |
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HO
0.87 |
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Morphological similarity: |
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Measure the molecules from the outside |
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Similarityrity betweenbetween moleculesules isis defineddefined asas aa functionon ofof thethe differencesdifferences in surfaceface measurementsmeasurements from observationbservation pointspoints..
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Data |
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Data from: G. Jones, P. Willett, R. C. Glen, A. R. Leach, & R. Taylor, J. Mol. Biol
267(1997) 727-748
♦134 protein/ligand complexes (> 20 different proteins with multiple ligands)
♦74 related pairs of molecules (small sample from space of all possible related pairs of molecules)
♦680 unrelated pairs (randomly selected set above, avoiding pairs known to bind competitively)
See: A. N. Jain. Morphological Similarity...
J. Comp.-Aided Mol. Design. 14: 199-213, 2000.
For each technique, we compute an estimate of two distributions
♦Distribution of random variable X (similarity function of ω, the pair of molecules) for ω in the space of related pairs
♦Distribution of random variable X (similarity function of ω, the pair of molecules) for ω in the space of unrelated pairs
♦Compare the estimated density functions and the cumulative distribution functions
UCSF |
Molecular similarity: 2D |
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2D similarity
♦Graph-based approach to comparing organic structures
♦Very efficient algorithm
♦Can search 100,000 compounds in seconds
What is the algorithm?
♦We compute all atomic paths of length K in a molecule of size N atoms
♦We mark a bit in a long bitstring if the corresponding path exists
♦We fold the bitstring in half many times, performing an OR, thus yielding a short bitstring
♦Given bitstrings A and B, we compute the number of bits in common divided by the total number of bits in either
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00..8989 |
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0.82 |
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0.73 |
00..6565 |
00..5858 |
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0.57 |
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0.54 |
00..4545 |
00..1313 |
Complexity: Computing the bitstring is O(N); computing S(A,B) is essentially constant time (small constant!)