Computational Biology

computational-biology-ai999x

DNA and protein sequences, alignment, molecular weights, genetics and population genetics, phylogenetics, enzyme kinetics, metabolic and population models, and ecology.

Life Sciences Question Answering Token Classification Text Classification Sentence Similarity Feature Extraction

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83capabilities

10topics

83formulas typeset

8worked examples below

1.0.0version

About this model

What Computational Biology does

Computational Biology (computational-biology-ai999x) is an offline Python package for computational biology. It covers DNA and protein sequences, alignment, molecular weights, genetics and population genetics, phylogenetics, enzyme kinetics, metabolic and population models, and ecology.

It has 83 capabilities — 83 computational procedures — in 10 topics, the largest being Sequences, Population genetics, Ecology, Enzyme kinetics, Alignment and Phylogenetics. Ask a question in words with ask(), or call a capability by name with typed inputs with solve(). Every answer carries its value and unit, the steps that produced it and the formula in LaTeX.

When a question is not determined — a value is missing, an input is out of range, or the request is outside the package — it says so and names what it needs, instead of guessing. It runs on an ordinary CPU without a network connection (Python >=3.10), and the same question always gets the same answer.

Install

pip install computational-biology-ai999x
hf download AI999X/computational-biology-ai999x --local-dir computational-biology-ai999x

What it covers

Alignment7Global alignment score (Needleman-Wunsch, linear gaps) · Global alignment score with affine gaps (Gotoh) · Global protein alignment score (BLOSUM62, affine gaps) …Ecology14ACE species richness estimator · Berger-Parker and McIntosh indices · Beta diversity between two samples …Enzyme kinetics9Competitive inhibition global fit · Eadie-Hofstee fit of Michaelis-Menten data · Hanes-Woolf fit of Michaelis-Menten data …Genetics5Inbreeding coefficient from a pedigree · Map distance from a recombination fraction · Phenotype ratio of a cross (complete dominance) …Microbial growth1Chemostat steady state (Monod kinetics)Phylogenetics7Jukes-Cantor distance · Kimura two-parameter distance · Neighbour-joining branch lengths …Population biology2Leslie matrix growth rate and stable age distribution · Leslie matrix population projection Population genetics16Allele frequency under haploid (genic) selection · Decline of a recessive lethal allele · Effective population size with unequal sex ratio …Sequences21Amino-acid composition of a protein · Aromaticity of a protein · Edit (Levenshtein) distance between sequences …Systems biology1Flux balance analysis (optimal objective)

Try it

Worked examples

Each question below was put to computational-biology-ai999x and the answer shown is the one it returned, with the steps it took.

Sequencesm.ask(…)

Reverse complement: seq = ATGCCGTA

revcomp = TACGGCAT

Show the steps

Reverse complement of a DNA sequence
5'\text{-}x_1\dots x_n\text{-}3' \mapsto 5'\text{-}\bar x_n\dots \bar x_1\text{-}3'
with seq = ATGCCGTA
→ revcomp = TACGGCAT

Capability: Reverse complement of a DNA sequence

Population geneticsm.ask(…)

Hardy-Weinberg test: n_AA = 298, n_Aa = 489, n_aa = 213

p = 0.5425

q = 0.4575

exp_AA = 294.30625

exp_Aa = 496.3875

exp_aa = 209.30625

chi2 = 0.2214895876

Show the steps

Hardy-Weinberg chi-square test
\chi^2 = \sum \frac{(O-E)^2}{E},\quad E = (Np^2,\ 2Npq,\ Nq^2)
with n_AA = 298, n_Aa = 489, n_aa = 213
→ p = 0.5425, q = 0.4575, exp_AA = 294.30625, exp_Aa = 496.3875, exp_aa = 209.30625, chi2 = 0.2214895876, p_value = 0.6379072638, Ho = 0.489, He = 0.4963875, F = 0.01488252625

Capability: Hardy-Weinberg chi-square test

Alignmentm.ask(…)

Global alignment score: seq1 = GATTACA, seq2 = GCATGCU, match = 1, mismatch = -1, gap = -1

score = 0

Show the steps

Global alignment score (Needleman-Wunsch, linear gaps)
S_{i,j} = \max(S_{i-1,j-1}+s(x_i,y_j),\ S_{i-1,j}+g,\ S_{i,j-1}+g)
with seq1 = GATTACA, seq2 = GCATGCU, match = 1, mismatch = -1, gap = -1
→ score = 0

Capability: Global alignment score (Needleman-Wunsch, linear gaps)

Geneticsm.ask(…)

Cross phenotype ratio: parent1 = AaBb, parent2 = AaBb

p_all_dominant = 0.5625

p_all_recessive = 0.0625

n_genotypes = 9

n_phenotypes = 4

ratio = 9:3:3:1

Show the steps

Phenotype ratio of a cross (complete dominance)
(\tfrac34 D + \tfrac14 r)^n \Rightarrow 9:3:3:1 \text{ for two loci}
with parent1 = AaBb, parent2 = AaBb
→ p_all_dominant = 0.5625, p_all_recessive = 0.0625, n_genotypes = 9, n_phenotypes = 4, ratio = 9:3:3:1

Capability: Phenotype ratio of a cross (complete dominance)

Enzyme kineticsm.solve(…)

m.solve("bio_michaelis_menten_fit", S="0.5, 1, 2, 5", v="2.2, 3.5, 5.1, 6.6")

Vmax = 8.500055206

Km = 1.400014367

Show the steps

Michaelis-Menten nonlinear least-squares fit
\min_{V_{\max},K_m} \sum_i \Big(v_i - \frac{V_{\max}S_i}{K_m+S_i}\Big)^2
with S = [0.5, 1, 2, 5], v = [2.2, 3.5, 5.1, 6.6]
→ Vmax = 8.500055206, Km = 1.400014367

Capability: Michaelis-Menten nonlinear least-squares fit

Ecologym.ask(…)

ACE richness estimator: counts = [1, 1, 2, 3, 5, 8, 20, 50], rare_threshold = 10

ace = 9.719298246

Show the steps

ACE species richness estimator
S_{ace} = S_{abund} + \frac{S_{rare}}{C_{ace}} + \frac{F_1}{C_{ace}}\gamma^2
with counts = [1, 1, 2, 3, 5, 8, 20, 50], rare_threshold = 10
→ ace = 9.719298246

Capability: ACE species richness estimator

Microbial growthm.ask(…)

Chemostat steady state: mu_max = 1, Ks = 1, Y = 0.5, D = 0.5, S0 = 10

S = 1

X = 4.5

Show the steps

Chemostat steady state (Monod kinetics)
\bar S = \frac{K_sD}{\mu_{\max} - D},\quad \bar X = Y(S_0 - \bar S)
with mu_max = 1, Ks = 1, Y = 0.5, D = 0.5, S0 = 10
→ S = 1, X = 4.5

Capability: Chemostat steady state (Monod kinetics)

Phylogeneticsm.ask(…)

Parsimony score: tree = ((A,B),(C,D));, states = A:AACG B:AATG C:GACT D:GGCT

score = 4

Show the steps

Parsimony score of a tree (Fitch)
c_v(a) = \sum_{w\in\mathrm{children}(v)} \min_b \big(c_w(b) + [a\ne b]\big)
with tree = ((A,B),(C,D));, states = A:AACG B:AATG C:GACT D:GGCT
→ score = 4

Capability: Parsimony score of a tree (Fitch)

Use it

From Python or the command line

Every snippet was run as shown; the output under it is what it printed.

import computational_biology_ai999x as m

a = m.ask("Reverse complement: seq = ATGCCGTA")
print(a.outcome)
print(a.values)
for step in a.steps:
    print(step["name"])

Output

SOLVED
{'revcomp': 'TACGGCAT'}
Reverse complement of a DNA sequence

import computational_biology_ai999x as m

a = m.solve("bio_michaelis_menten_fit", S="0.5, 1, 2, 5", v="2.2, 3.5, 5.1, 6.6")
print(a.outcome)
print(a.values)

Output

SOLVED
{'Vmax': 8.50005520631, 'Km': 1.40001436689}

import computational_biology_ai999x as m

# an input is missing: the answer says what it needs instead of guessing
a = m.solve("bio_michaelis_menten_fit")
print(a.outcome)
print(a.message)

Output

NEEDS_INPUT
bio_michaelis_menten_fit needs S, v (substrate concentrations (any one unit); initial rates (any one unit)). Example: solve('bio_michaelis_menten_fit', S=[1, 2, 4, 8], v=[5, 6.6666666667, 8, 8.8888888889]).

computational-biology-ai999x "Reverse complement: seq = ATGCCGTA"

Output

revcomp = TACGGCAT
1. Reverse complement of a DNA sequence: complement each base (A-T, C-G) and reverse the order  with seq = ATGCCGTA  ->  revcomp = TACGGCAT
Assumptions: DNA letters A, C, G, T only
…

Figures

Computed, not drawn by hand

Every value in these figures was computed by the package, each point a run of solve(); in a notebook, plot() and plot_relation() draw a capability's curves the same way.

Michaelis–Menten kinetics: measured rates and fitted curve

\min_{V_{\max},K_m} \sum_i \Big(v_i - \frac{V_{\max}S_i}{K_m+S_i}\Big)^2

Michaelis–Menten fit to 4 measured rates (S = [1, 2, 4, 8]): the package returns Vmax = 10 and Km = 1; the curve drawn is v = Vmax·S/(Km + S) with those two values (the double-reciprocal fit gives Vmax = 10, Km = 1, r² = 1). Marked: S = Km = 1, v = Vmax/2 = 5.

Enzyme progress curve: substrate and product over time

K_m\ln\frac{S_0}{S} + (S_0 - S) = V_{\max}t

Substrate and product over time for an enzyme reaction with S0 = 10, Vmax = 1 and Km = 2 (integrated Michaelis–Menten law), computed by the package at 101 times. Half the substrate is converted at t = 6.38629; marked: t = 6.38629, S = 5.

Chemostat steady states against dilution rate

\bar S = \frac{K_sD}{\mu_{\max} - D},\quad \bar X = Y(S_0 - \bar S)

Chemostat operating diagram (Monod growth, μmax = 1, Ks = 1, Y = 0.5, feed S0 = 10): steady-state substrate and biomass computed by the package at 90 dilution rates up to washout; biomass productivity D·X peaks near D = 0.7 (2.68333). Marked: D = 0.5, X = 4.5, S = 1.

Age-structured population growth (Leslie matrix)

\mathbf n_{t+1} = L\,\mathbf n_t,\quad L = \begin{pmatrix}F_1 & F_2 & \cdots\\ S_1 & 0 & \\ & \ddots & \end{pmatrix}

Leslie-matrix projection of an age-structured population (fecundities F = [0, 1.5, 1], survival S = [0.5, 0.8], start n0 = [100, 50, 20]): each bar is the population the package returns at that step, split by age class. Growth settles to λ = 1.06152 per step (r = 0.0596993); at t = 12 the classes are [197.307, 92.9426, 69.9967]; marked: total = 360.247.

Hardy–Weinberg: observed against expected genotypes

\chi^2 = \sum \frac{(O-E)^2}{E},\quad E = (Np^2,\ 2Npq,\ Nq^2)

Observed genotype counts AA = 298, Aa = 489, aa = 213 beside the counts expected under Hardy–Weinberg equilibrium, computed by the package from the allele frequencies p = 0.5425, q = 0.4575: 294.306, 496.387, 209.306. χ² = 0.22149, p-value 0.637907; observed heterozygosity 0.489 against expected 0.496387 (F = 0.0148825).

Heterozygosity lost by drift: population size against time

H_t = H_0\left(1 - \frac{1}{2N}\right)^t

Fraction of heterozygosity retained under random genetic drift, (1 − 1/(2N))^t, for population sizes N = 5, 10, 25, 50, 100, 250 (rows) after 10, 25, 50, 100, 200 generations (columns), each cell computed by the package and rounded to 3 decimals. N = 50 keeps 0.366 after 100 generations; N = 5 keeps 0 after 200.

Formulas

The mathematics inside

12 of its 83 formulas; every one is in the capability browser below.

Reverse complement of a DNA sequence

5'\text{-}x_1\dots x_n\text{-}3' \mapsto 5'\text{-}\bar x_n\dots \bar x_1\text{-}3'

Hardy-Weinberg chi-square test

\chi^2 = \sum \frac{(O-E)^2}{E},\quad E = (Np^2,\ 2Npq,\ Nq^2)

Global alignment score (Needleman-Wunsch, linear gaps)

S_{i,j} = \max(S_{i-1,j-1}+s(x_i,y_j),\ S_{i-1,j}+g,\ S_{i,j-1}+g)

Phenotype ratio of a cross (complete dominance)

(\tfrac34 D + \tfrac14 r)^n \Rightarrow 9:3:3:1 \text{ for two loci}

Michaelis-Menten nonlinear least-squares fit

\min_{V_{\max},K_m} \sum_i \Big(v_i - \frac{V_{\max}S_i}{K_m+S_i}\Big)^2

ACE species richness estimator

S_{ace} = S_{abund} + \frac{S_{rare}}{C_{ace}} + \frac{F_1}{C_{ace}}\gamma^2

Chemostat steady state (Monod kinetics)

\bar S = \frac{K_sD}{\mu_{\max} - D},\quad \bar X = Y(S_0 - \bar S)

Parsimony score of a tree (Fitch)

c_v(a) = \sum_{w\in\mathrm{children}(v)} \min_b \big(c_w(b) + [a\ne b]\big)

Global alignment score with affine gaps (Gotoh)

g(L) = o + (L-1)\,e

Berger-Parker and McIntosh indices

d = \frac{n_{\max}}{N},\quad D = \frac{N - U}{N - \sqrt N}

Competitive inhibition global fit

v = \frac{V_{\max}S}{K_m\left(1 + \frac{[I]}{K_i}\right) + S}

Inbreeding coefficient from a pedigree

F_X = \sum_{\text{paths}} \left(\tfrac12\right)^{n_1+n_2+1}(1 + F_A)

Capabilities

Every capability, with an example

Search by name, formula, input or topic. Each capability opens to show what it takes, what it returns, when it applies, and a question it answered.

83 capabilities

Alignment 7

Global alignment score (Needleman-Wunsch, linear gaps)bio_global_alignment_score

The optimal global (Needleman-Wunsch) alignment score of two sequences with stated match, mismatch and linear gap scores.

S_{i,j} = \max(S_{i-1,j-1}+s(x_i,y_j),\ S_{i-1,j}+g,\ S_{i,j-1}+g)

Inputs

seq1 first sequence text
seq2 second sequence text
match score of a match
mismatch score of a mismatch
gap score of each gap position (<= 0)

Returns

score optimal global alignment score

Applies when

gap <= 0
end gaps scored like internal gaps

Example

m.ask("Global alignment score: seq1 = GATTACA, seq2 = GCATGCU, match = 1, mismatch = -1, gap = -1")

→ score = 0

Global alignment score with affine gaps (Gotoh)bio_global_alignment_affine

The optimal global alignment score with affine gaps: a gap of length L scores open + (L-1)*extend.

g(L) = o + (L-1)\,e

Inputs

seq1 first sequence text
seq2 second sequence text
match score of a match
mismatch score of a mismatch
open score of the first position of a gap (<= 0)
extend score of each further gap position (<= 0)

Returns

score optimal global alignment score

Applies when

open, extend <= 0
a gap of length L scores open + (L-1) extend
end gaps scored

Example

m.ask("Global alignment score with affine gaps: seq1 = ACGTTTACG, seq2 = ACGACG, match = 2, mismatch = -1, open = -5, extend = -1")

→ score = 5

Global protein alignment score (BLOSUM62, affine gaps)bio_protein_global_blosum62

The optimal global alignment score of two proteins with BLOSUM62 and affine gaps.

S = \max_{\text{alignments}} \sum B_{62}(x,y) + \sum_{\text{gaps}} [o + (L-1)e]

Inputs

seq1 first protein sequence text
seq2 second protein sequence text
open score of the first position of a gap (<= 0)
extend score of each further gap position (<= 0)

Returns

score optimal global alignment score

Applies when

standard amino acids
open, extend <= 0

Example

m.ask("Global protein alignment score: seq1 = HEAGAWGHEE, seq2 = PAWHEAE, open = -10, extend = -1")

→ score = 3

Local alignment score (Smith-Waterman, linear gaps)bio_local_alignment_score

The optimal local (Smith-Waterman) alignment score of two sequences with stated match, mismatch and linear gap scores.

S_{i,j} = \max(0,\ S_{i-1,j-1}+s(x_i,y_j),\ S_{i-1,j}+g,\ S_{i,j-1}+g)

Inputs

seq1 first sequence text
seq2 second sequence text
match score of a match
mismatch score of a mismatch
gap score of each gap position (<= 0)

Returns

score optimal local alignment score (>= 0)

Applies when

match > 0
gap <= 0

Example

m.ask("Local alignment score: seq1 = TGTTACGG, seq2 = GGTTGACTA, match = 3, mismatch = -3, gap = -2")

→ score = 13

Local alignment score with affine gapsbio_local_alignment_affine

The optimal local alignment score with affine gaps (a gap of length L scores open + (L-1)*extend).

g(L) = o + (L-1)\,e,\quad S \ge 0

Inputs

seq1 first sequence text
seq2 second sequence text
match score of a match
mismatch score of a mismatch
open score of the first position of a gap (<= 0)
extend score of each further gap position (<= 0)

Returns

score optimal local alignment score (>= 0)

Applies when

match > 0
open, extend <= 0

Example

m.ask("Local alignment score with affine gaps: seq1 = TTACGTACGTTT, seq2 = GGACGTTACGTGG, match = 2, mismatch = -1, open = -3, extend = -1")

→ score = 13

Local protein alignment score (BLOSUM62, affine gaps)bio_protein_local_blosum62

The optimal local (Smith-Waterman) alignment score of two proteins with BLOSUM62 and affine gaps.

S = \max\Big(0, \max_{\text{local alignments}} \textstyle\sum B_{62} + \sum [o + (L-1)e]\Big)

Inputs

seq1 first protein sequence text
seq2 second protein sequence text
open score of the first position of a gap (<= 0)
extend score of each further gap position (<= 0)

Returns

score optimal local alignment score (>= 0)

Applies when

standard amino acids
open, extend <= 0

Example

m.ask("Local protein alignment score: seq1 = HEAGAWGHEE, seq2 = PAWHEAE, open = -8, extend = -8")

→ score = 20

Semi-global (end-gap-free) alignment scorebio_semiglobal_alignment_score

The optimal alignment score when gaps at either end of either sequence are free (fitting a read inside a longer sequence, overlaps), with linear internal gaps.

S = \max\big(\max_j S_{n,j},\ \max_i S_{i,m}\big),\ S_{i,0}=S_{0,j}=0

Inputs

seq1 first sequence text
seq2 second sequence text
match score of a match
mismatch score of a mismatch
gap score of each internal gap position (<= 0)

Returns

score optimal semi-global alignment score

Applies when

gap <= 0
end gaps unscored

Example

m.ask("Semi-global alignment score: seq1 = TTTTACGTACGGGG, seq2 = ACGTAC, match = 1, mismatch = -1, gap = -1")

→ score = 6

Ecology 14

ACE species richness estimatorbio_ace

The abundance-based coverage estimator (ACE) of species richness, with a stated rare-species threshold (10 is the usual value).

S_{ace} = S_{abund} + \frac{S_{rare}}{C_{ace}} + \frac{F_1}{C_{ace}}\gamma^2

Inputs

counts number of individuals of each species vector
rare_threshold largest count of a rare species (usually 10) integer

Returns

ace estimated species richness

Applies when

whole counts
not every rare species a singleton

Example

m.ask("ACE richness estimator: counts = [1, 1, 2, 3, 5, 8, 20, 50], rare_threshold = 10")

→ ace = 9.7193

Berger-Parker and McIntosh indicesbio_dominance_indices

The Berger-Parker dominance (share of the most abundant species) and McIntosh's index of a sample.

d = \frac{n_{\max}}{N},\quad D = \frac{N - U}{N - \sqrt N}

Inputs

counts number of individuals of each species vector

Returns

berger_parker Berger-Parker dominance n_max/N
mcintosh McIntosh index (N - U)/(N - sqrt N), U = sqrt(sum n_i^2)

Applies when

whole counts
N >= 2

Example

m.ask("Berger-Parker and McIntosh indices: counts = [50, 30, 20]")

→ berger_parker = 0.5, mcintosh = 0.426176

Beta diversity between two samplesbio_beta_diversity

The Bray-Curtis dissimilarity (abundances) and the Jaccard and Sorensen dissimilarities (presence and absence) between two community samples.

BC = \frac{\sum|x_i - y_i|}{\sum (x_i + y_i)},\quad J = 1 - \frac{|A\cap B|}{|A\cup B|}

Inputs

x abundances of each species in the first sample vector
y abundances of the same species in the second sample vector

Returns

bray_curtis Bray-Curtis dissimilarity
jaccard Jaccard dissimilarity (1 - shared/union)
sorensen Sorensen dissimilarity (1 - 2 shared/(S_x + S_y))

Applies when

the same species order in both samples
each sample non-empty

Example

m.ask("Beta diversity: x = [6, 7, 4, 0], y = [10, 0, 6, 3]")

→ bray_curtis = 0.444444, jaccard = 0.5, sorensen = 0.333333

Brillouin diversity indexbio_brillouin_diversity

Brillouin's diversity index of a fully censused collection (natural logarithm).

H_B = \frac{\ln N! - \sum_i \ln n_i!}{N}

Inputs

counts number of individuals of each species vector

Returns

H_B Brillouin index (nats per individual)

Applies when

whole counts
N <= 20000

Example

m.ask("Brillouin diversity: counts = [5, 3, 2]")

→ H_B = 0.783201

Chao1 species richness estimatorbio_chao1

The Chao1 estimate of total species richness from abundance counts (bias-corrected or classic form), with the observed richness and the numbers of singletons and doubletons.

\hat S = S_{obs} + \frac{F_1(F_1-1)}{2(F_2+1)}

Inputs

counts number of individuals of each species vector
form bias_corrected or classic Chao1 choice

Returns

chao1 estimated species richness
s_obs observed number of species
singletons species seen once (F1)
doubletons species seen twice (F2)

Applies when

whole counts
the classic form needs F1 > 0 and F2 > 0, otherwise the bias-corrected form is used (as scikit-bio does)

Example

m.ask("Chao1: counts = [1, 1, 1, 2, 2, 5, 10], form = classic")

→ chao1 = 9.25, s_obs = 7, singletons = 3, doubletons = 2

Faith's phylogenetic diversitybio_faith_pd

Faith's phylogenetic diversity of a set of observed taxa: the total branch length of the rooted tree connecting them to the root.

PD = \sum_{b \in T(\text{taxa}) \cup R} \ell_b

Inputs

tree rooted tree in Newick format with branch lengths text
taxa observed taxa (leaf names separated by spaces) text

Returns

pd Faith's PD (sum of branch lengths)

Applies when

every branch has a length
taxa are leaves of the tree
rooted convention (path to the root included)

Example

m.ask("Faith's phylogenetic diversity: tree = ((A:1,B:2):3,(C:4,D:5):6);, taxa = A B")

→ pd = 6

Good's coverage of a samplebio_goods_coverage

Good's coverage estimator: the estimated fraction of the community's individuals belonging to species already seen in the sample.

C = 1 - \frac{F_1}{N}

Inputs

counts number of individuals of each species vector

Returns

coverage Good's coverage 1 - F1/N

Applies when

whole counts

Example

m.ask("Good's coverage: counts = [1, 1, 2, 6]")

→ coverage = 0.8

Margalef and Menhinick richness indicesbio_richness_indices

Margalef's richness index (S - 1)/ln N and Menhinick's richness index S/sqrt(N) of a sample.

D_{Mg} = \frac{S-1}{\ln N},\quad D_{Mn} = \frac{S}{\sqrt N}

Inputs

counts number of individuals of each species vector

Returns

margalef Margalef's index (S - 1)/ln N
menhinick Menhinick's index S/sqrt(N)

Applies when

whole counts
N >= 2

Example

m.ask("Richness indices: counts = [10, 20, 30, 40]")

→ margalef = 0.651442, menhinick = 0.4

Mark-recapture population estimate (Petersen MLE)bio_petersen_estimate

The maximum-likelihood population size from one mark-recapture experiment: M animals marked, C caught later, R of them marked (hypergeometric likelihood).

\hat N = \left\lfloor \frac{MC}{R} \right\rfloor

Inputs

M number marked and released integer
C number caught in the second sample integer
R number of marked animals recaptured integer

Returns

N_mle maximum-likelihood population size (largest maximiser)

Applies when

closed population
R >= 1, R <= min(M, C)
when MC/R is whole, N = MC/R and MC/R - 1 tie; the larger is given

Example

m.ask("Petersen estimate: M = 100, C = 80, R = 12")

→ N_mle = 666

Rarefied species richnessbio_rarefaction

The expected number of species in a random subsample of n individuals drawn without replacement (Hurlbert-Sanders rarefaction).

E[S_n] = \sum_i \left[1 - \binom{N-N_i}{n}\Big/\binom{N}{n}\right]

Inputs

counts number of individuals of each species vector
n subsample size (individuals) integer

Returns

expected_species expected number of species in the subsample

Applies when

whole counts
1 <= n <= N

Example

m.ask("Rarefaction: counts = [2, 1, 1], n = 2")

→ expected_species = 1.83333

Schumacher-Eschmeyer multiple-census estimatebio_schumacher_eschmeyer

The Schumacher-Eschmeyer estimate of a closed population's size from repeated mark-recapture samples.

\hat N = \frac{\sum_t C_tM_t^2}{\sum_t R_tM_t}

Inputs

C number caught on each occasion vector
R number of those already marked vector
M number of marked animals at large before each occasion vector

Returns

N estimated population size

Applies when

closed population
whole counts, R <= C
at least one recapture of a marked animal

Example

m.ask("Schumacher-Eschmeyer estimate: C = [10, 10, 10], R = [0, 2, 3], M = [0, 10, 18]")

→ N = 57.2973

Shannon diversity indexbio_shannon_diversity

The Shannon diversity index H' of a community in the stated logarithm base, Pielou's evenness J' and the effective number of species exp(H').

H' = -\sum_i p_i \log_b p_i,\quad J' = \frac{H'}{\log_b S}

Inputs

counts number of individuals of each species vector
base logarithm base: e (nats), 2 (bits) or 10 choice

Returns

H Shannon index H' in the stated base
evenness Pielou's evenness J' = H'/ln S
effective_number effective number of species exp(H') (Hill number of order 1)

Applies when

whole non-negative counts
zero counts are not species of the sample

Example

m.ask("Shannon diversity: counts = [10, 10, 10, 10], base = e")

→ H = 1.38629, evenness = 1, effective_number = 4

Simpson diversity indicesbio_simpson_diversity

Simpson's dominance D, Simpson's diversity 1 - D and the inverse Simpson index 1/D, with the stated estimator (infinite population: sum p^2; finite sample: sum n(n-1)/(N(N-1))).

D = \sum_i p_i^2 \quad\text{or}\quad D = \frac{\sum n_i(n_i-1)}{N(N-1)}

Inputs

counts number of individuals of each species vector
estimator infinite (sum p^2) or finite-sample correction choice

Returns

D Simpson's dominance D
simpson Simpson's diversity 1 - D
inverse inverse Simpson 1/D

Applies when

whole non-negative counts
estimator stated

Example

m.ask("Simpson diversity: counts = [10, 10, 10, 10], estimator = infinite")

→ D = 0.25, simpson = 0.75, inverse = 4

Species-area relationship fitbio_species_area

The constants c and z of the Arrhenius species-area relationship S = c A^z fitted by least squares on log S against log A.

\ln S = \ln c + z\ln A

Inputs

A areas (any one unit) vector
S species counts vector

Returns

c constant c (species in a unit area)
z exponent z
r2 r^2 of the log-log fit

Applies when

positive areas and counts
c refers to the area unit used

Example

m.ask("Species-area relationship: A = [1, 10, 100, 1000], S = [10, 20, 40, 80]")

→ c = 10, z = 0.30103, r2 = 1

Enzyme kinetics 9

Competitive inhibition global fitbio_competitive_inhibition_fit

Vmax, Km and the inhibition constant Ki of a competitive inhibitor fitted globally to rates measured at several substrate and inhibitor concentrations (nonlinear least squares, unweighted).

v = \frac{V_{\max}S}{K_m\left(1 + \frac{[I]}{K_i}\right) + S}

Inputs

S substrate concentrations (one per measurement) vector
I inhibitor concentrations (one per measurement, 0 for none) vector
v measured rates vector

Returns

Vmax maximum rate
Km Michaelis constant (unit of S)
Ki inhibition constant (unit of I)

Applies when

rates without inhibitor at 3 or more S
rates with inhibitor
convergence

Example

m.ask("Competitive inhibition fit: S = [1, 2, 4, 8, 1, 2, 4, 8], I = [0, 0, 0, 0, 2, 2, 2, 2], v = [5, 6.66666666667, 8, 8.88888888889, 2.5, 4, 5.71428571429, 7.27272727273]")

→ Vmax = 10, Km = 1, Ki = 1

Eadie-Hofstee fit of Michaelis-Menten databio_eadie_hofstee

Vmax and Km from a least-squares line through the Eadie-Hofstee plot of v against v/S.

v = V_{\max} - K_m\frac{v}{S}

Inputs

S substrate concentrations (any one unit) vector
v initial rates (any one unit) vector

Returns

Vmax maximum rate (unit of v)
Km Michaelis constant (unit of S)
r2 coefficient of determination of the linear fit

Applies when

positive S and v
negative slope, positive intercept

Example

m.ask("Eadie-Hofstee fit: S = [1, 2, 4, 8], v = [5, 6.6666666667, 8, 8.8888888889]")

→ Vmax = 10, Km = 1, r2 = 1

Hanes-Woolf fit of Michaelis-Menten databio_hanes_woolf

Vmax and Km from a least-squares line through the Hanes-Woolf plot of S/v against S.

\frac Sv = \frac{1}{V_{\max}}S + \frac{K_m}{V_{\max}}

Inputs

S substrate concentrations (any one unit) vector
v initial rates (any one unit) vector

Returns

Vmax maximum rate (unit of v)
Km Michaelis constant (unit of S)
r2 coefficient of determination of the linear fit

Applies when

positive S and v
positive slope and intercept

Example

m.ask("Hanes-Woolf fit: S = [1, 2, 4, 8], v = [5, 6.6666666667, 8, 8.8888888889]")

→ Vmax = 10, Km = 1, r2 = 1

Hill coefficient from a Hill plotbio_hill_plot

The Hill coefficient n and half-saturation constant K from a least-squares line through the Hill plot of log10(Y/(1-Y)) against log10 S (Y = fractional saturation).

\log\frac{Y}{1-Y} = n\log S - n\log K

Inputs

S ligand or substrate concentrations vector
Y fractional saturations (0 < Y < 1) vector

Returns

n Hill coefficient
K half-saturation concentration (unit of S)
r2 coefficient of determination of the line

Applies when

0 < Y < 1
positive S
rising Hill plot

Example

m.ask("Hill plot: S = [1, 2, 4, 8], Y = [0.0588235294118, 0.2, 0.5, 0.8]")

→ n = 2, K = 4, r2 = 1

Hill equation nonlinear least-squares fitbio_hill_fit

Vmax, the half-saturation constant K and the Hill coefficient n fitted to rate or binding data by nonlinear least squares (unweighted).

v = \frac{V_{\max}S^n}{K^n + S^n}

Inputs

S concentrations (any one unit) vector
v rates or responses (any one unit) vector

Returns

Vmax maximum rate
K half-saturation concentration (unit of S)
n Hill coefficient

Applies when

at least 4 distinct positive concentrations
positive responses
convergence

Example

m.ask("Hill fit: S = [1, 2, 4, 8, 16], v = [0.588235294118, 2, 5, 8, 9.41176470588]")

→ Vmax = 10, K = 4, n = 2

Lineweaver-Burk fit of Michaelis-Menten databio_lineweaver_burk

Vmax and Km from a least-squares line through the double-reciprocal (Lineweaver-Burk) plot of 1/v against 1/S.

\frac1v = \frac{K_m}{V_{\max}}\frac1S + \frac1{V_{\max}}

Inputs

S substrate concentrations (any one unit) vector
v initial rates (any one unit) vector

Returns

Vmax maximum rate (unit of v)
Km Michaelis constant (unit of S)
r2 coefficient of determination of the linear fit

Applies when

positive S and v
positive slope and intercept

Example

m.ask("Lineweaver-Burk fit: S = [1, 2, 4, 8], v = [5, 6.6666666667, 8, 8.8888888889]")

→ Vmax = 10, Km = 1, r2 = 1

Michaelis-Menten nonlinear least-squares fitbio_michaelis_menten_fit

Vmax and Km of the Michaelis-Menten equation fitted to rate data by nonlinear least squares (unweighted).

\min_{V_{\max},K_m} \sum_i \Big(v_i - \frac{V_{\max}S_i}{K_m+S_i}\Big)^2

Inputs

S substrate concentrations (any one unit) vector
v initial rates (any one unit) vector

Returns

Vmax maximum rate (unit of v)
Km Michaelis constant (unit of S)

Applies when

at least 3 distinct positive S
positive v
an interior least-squares optimum

Example

m.ask("Michaelis-Menten fit: S = [1, 2, 4, 8], v = [5, 6.6666666667, 8, 8.8888888889]")

→ Vmax = 10, Km = 1

Substrate remaining in a Michaelis-Menten reactionbio_michaelis_menten_progress

The substrate and product concentrations after a time t of an irreversible Michaelis-Menten reaction (integrated rate law; constant enzyme, no product inhibition).

K_m\ln\frac{S_0}{S} + (S_0 - S) = V_{\max}t

Inputs

S0 initial substrate concentration
Vmax maximum rate (concentration per time)
Km Michaelis constant (concentration)
t elapsed time (time unit of Vmax)

Returns

S substrate concentration at t
P product concentration at t

Applies when

positive S0, Vmax, Km
t >= 0
consistent units

Example

m.ask("Michaelis-Menten progress curve: S0 = 10, Vmax = 1, Km = 2, t = 5")

→ S = 6.01624, P = 3.98376

Time to reach a conversion in a Michaelis-Menten reactionbio_michaelis_menten_time

The time an irreversible Michaelis-Menten reaction needs to convert a stated fraction of its substrate.

t = \frac{K_m\ln\frac{1}{1-X} + S_0X}{V_{\max}}

Inputs

S0 initial substrate concentration
Vmax maximum rate (concentration per time)
Km Michaelis constant (concentration)
X fraction of substrate converted (0 to 1, or a percentage)

Returns

t time (time unit of Vmax)

Applies when

0 < X < 1
positive S0, Vmax, Km

Example

m.ask("Michaelis-Menten time to conversion: S0 = 10, Vmax = 1, Km = 2, X = 0.5")

→ t = 6.38629

Genetics 5

Inbreeding coefficient from a pedigreebio_pedigree_inbreeding

Wright's inbreeding coefficient F of an individual from its pedigree (founders unrelated and not inbred).

F_X = \sum_{\text{paths}} \left(\tfrac12\right)^{n_1+n_2+1}(1 + F_A)

Inputs

pedigree parent-child pairs, e.g. [P-S, Q-S, P-D, Q-D, S-X, D-X] edges
individual name of the individual text

Returns

F inbreeding coefficient

Applies when

every non-founder has exactly two parents listed
no cycles
founders unrelated, F = 0

Example

m.ask("Pedigree inbreeding coefficient: pedigree = [P-S, Q-S, P-D, Q-D, S-X, D-X], individual = X")

→ F = 0.25

Map distance from a recombination fractionbio_map_distance

The genetic map distance (cM) corresponding to a recombination fraction, by the Haldane or Kosambi map function.

d_H = -\tfrac12\ln(1-2r),\quad d_K = \tfrac14\ln\frac{1+2r}{1-2r}

Inputs

r recombination fraction (0 to 0.5)
map map function: haldane (no interference) or kosambi choice

Returns

d_cM map distance in centimorgans

Applies when

0 <= r < 0.5
map function stated

Example

m.ask("Map distance: r = 0.2, map = haldane")

→ d_cM = 25.5413

Phenotype ratio of a cross (complete dominance)bio_cross_phenotype_ratio

The offspring phenotype ratio of a cross with complete dominance, the probabilities of the all-dominant and all-recessive phenotypes and the numbers of genotypes and phenotypes.

(\tfrac34 D + \tfrac14 r)^n \Rightarrow 9:3:3:1 \text{ for two loci}

Inputs

parent1 genotype of the first parent, letter pairs (e.g. AaBb) text
parent2 genotype of the second parent text

Returns

p_all_dominant probability of showing every dominant trait
p_all_recessive probability of showing every recessive trait
n_genotypes number of distinct offspring genotypes
n_phenotypes number of distinct offspring phenotypes
ratio phenotype ratio, largest class first (e.g. 9:3:3:1)

Applies when

complete dominance at every locus
independent assortment

Example

m.ask("Cross phenotype ratio: parent1 = AaBb, parent2 = AaBb")

→ p_all_dominant = 0.5625, p_all_recessive = 0.0625, n_genotypes = 9, n_phenotypes = 4, ratio = 9:3:3:1

Probability of an offspring genotype in a crossbio_cross_genotype_probability

The probability that an offspring of two parents has a stated genotype (monohybrid, dihybrid or more loci; independent assortment).

P = \prod_{\ell} P_\ell(g_\ell)

Inputs

parent1 genotype of the first parent, letter pairs (e.g. AaBb) text
parent2 genotype of the second parent text
offspring offspring genotype asked about text

Returns

probability probability of the offspring genotype
fraction the same as a fraction

Applies when

the same loci in the same order
independent assortment
at most 6 loci

Example

m.ask("Cross genotype probability: parent1 = Aa, parent2 = Aa, offspring = Aa")

→ probability = 0.5, fraction = 1/2

Recombination fraction from a map distancebio_recombination_fraction

The recombination fraction between two loci a stated map distance apart, by the Haldane or Kosambi map function.

r_H = \tfrac12(1 - e^{-2d}),\quad r_K = \tfrac12\tanh 2d

Inputs

d_cM map distance in centimorgans
map map function: haldane (no interference) or kosambi choice

Returns

r recombination fraction

Applies when

d >= 0
map function stated

Example

m.ask("Recombination fraction: d_cM = 50, map = haldane")

→ r = 0.31606

Microbial growth 1

Chemostat steady state (Monod kinetics)bio_chemostat_steady_state

The steady-state substrate and biomass concentrations of a chemostat with Monod growth kinetics.

\bar S = \frac{K_sD}{\mu_{\max} - D},\quad \bar X = Y(S_0 - \bar S)

Inputs

mu_max maximum specific growth rate (1/time)
Ks half-saturation constant (concentration)
Y biomass yield (biomass per substrate)
D dilution rate (1/time, same unit as mu_max)
S0 feed substrate concentration

Returns

S steady-state substrate concentration
X steady-state biomass concentration

Applies when

D < mu_max
no washout (S < S0)
consistent units

Example

m.ask("Chemostat steady state: mu_max = 1, Ks = 1, Y = 0.5, D = 0.5, S0 = 10")

→ S = 1, X = 4.5

Phylogenetics 7

Jukes-Cantor distancebio_jukes_cantor_distance

The Jukes-Cantor (JC69) evolutionary distance between two aligned DNA sequences: the expected number of substitutions per site, corrected for multiple hits.

d = -\tfrac34 \ln\!\left(1 - \tfrac43 p\right)

Inputs

seq1 first aligned DNA sequence text
seq2 second aligned DNA sequence text

Returns

p proportion of differing sites
d Jukes-Cantor distance (substitutions per site)

Applies when

equal lengths, no gaps
p < 0.75

Example

m.ask("Jukes-Cantor distance: seq1 = AAAAAAAAAA, seq2 = AAAAAAAACC")

→ p = 0.2, d = 0.232616

Kimura two-parameter distancebio_kimura_distance

The Kimura two-parameter (K80) distance between two aligned DNA sequences, separating transitions (A<->G, C<->T) from transversions.

d = -\tfrac12\ln(1-2P-Q) - \tfrac14\ln(1-2Q)

Inputs

seq1 first aligned DNA sequence text
seq2 second aligned DNA sequence text

Returns

d K80 distance (substitutions per site)

Applies when

equal lengths, no gaps
1 - 2P - Q > 0 and 1 - 2Q > 0

Example

m.ask("Kimura distance: seq1 = AAAAAAAAAA, seq2 = AAAAAAAGGC")

→ d = 0.402359

Neighbour-joining branch lengthsbio_neighbor_joining

The total tree length and the terminal branch lengths of the neighbour-joining tree of a distance matrix.

Q_{ij} = (r-2)d_{ij} - R_i - R_j,\quad \ell_i = \tfrac12 d_{ij} + \frac{R_i - R_j}{2(r-2)}

Inputs

D symmetric distance matrix, zero diagonal (at least 3 taxa) matrix

Returns

total_length sum of all branch lengths
branch terminal branch length of each taxon, in input order

Applies when

at least 3 taxa
no ties in the joining criterion
negative branch lengths kept as computed

Example

m.ask("Neighbour joining: D = [[0, 5, 9, 9, 8], [5, 0, 10, 10, 9], [9, 10, 0, 8, 7], [9, 10, 8, 0, 3], [8, 9, 7, 3, 0]]")

→ total_length = 17, branch = [2.0, 3.0, 4.0, 2.0, 1.0]

Number of possible phylogenetic treesbio_tree_count

The number of distinct labelled bifurcating trees for n taxa, rooted or unrooted.

N_{\text{rooted}} = (2n-3)!!,\quad N_{\text{unrooted}} = (2n-5)!!

Inputs

n number of taxa integer
rooted rooted or unrooted trees choice

Returns

count number of distinct bifurcating trees

Applies when

n >= 2 (rooted) or n >= 3 (unrooted)
n <= 100

Example

m.ask("Number of trees: n = 5, rooted = unrooted")

→ count = 15

Parsimony score of a tree (Fitch)bio_fitch_parsimony

The minimum number of character changes (small parsimony score) needed on a given rooted bifurcating tree to explain aligned sequences at its leaves.

c_v(a) = \sum_{w\in\mathrm{children}(v)} \min_b \big(c_w(b) + [a\ne b]\big)

Inputs

tree rooted bifurcating tree in Newick format text
states aligned sequence of each leaf, written name:SEQUENCE separated by spaces text

Returns

score parsimony score (number of changes)

Applies when

strictly bifurcating rooted tree
one aligned sequence per leaf
unit cost per change

Example

m.ask("Parsimony score: tree = ((A,B),(C,D));, states = A:AACG B:AATG C:GACT D:GGCT")

→ score = 4

Robinson-Foulds distance between treesbio_robinson_foulds

The Robinson-Foulds (symmetric-difference) distance between two trees on the same taxa, comparing their bipartitions (trees treated as unrooted), and the distance as a proportion.

\mathrm{RF} = |S_1 \setminus S_2| + |S_2 \setminus S_1|

Inputs

tree1 first tree in Newick format text
tree2 second tree in Newick format text

Returns

rf number of bipartitions found in only one tree
rf_normalized RF divided by the total number of non-trivial bipartitions of both trees

Applies when

the same taxa
at least 4 taxa
trees compared unrooted; branch lengths ignored

Example

m.ask("Robinson-Foulds distance: tree1 = ((A,B),(C,D),(E,F));, tree2 = ((A,C),(B,D),(E,F));")

→ rf = 4, rf_normalized = 0.666667

UPGMA tree heightsbio_upgma

The node heights of the UPGMA (average-linkage) tree of a distance matrix: the root height and every merge height (half the distance between the merged clusters).

d_{(ij),k} = \frac{n_i d_{ik} + n_j d_{jk}}{n_i + n_j},\quad h = d_{ij}/2

Inputs

D symmetric distance matrix, zero diagonal matrix

Returns

root_height height of the root
heights all node heights, ascending

Applies when

symmetric, zero diagonal, positive distances
no ties in a merge step

Example

m.ask("UPGMA: D = [[0, 2, 4, 6], [2, 0, 4, 6], [4, 4, 0, 6], [6, 6, 6, 0]]")

→ root_height = 3, heights = [1.0, 2.0, 3.0]

Population biology 2

Leslie matrix growth rate and stable age distributionbio_leslie_growth_rate

The asymptotic growth rate (dominant eigenvalue lambda and r = ln lambda), the net reproductive rate R0 and the stable age distribution of a Leslie model.

\sum_{i} F_i\,l_i\,\lambda^{-i} = 1,\quad R_0 = \sum_i F_i l_i

Inputs

F fecundity of each age class vector
S survival from each age class to the next (one fewer than F) vector

Returns

lambda1 dominant eigenvalue (growth factor per step)
r intrinsic rate ln(lambda)
R0 net reproductive rate (expected offspring per newborn)
stable stable age distribution (fractions summing to 1)

Applies when

0 < S_i <= 1
F_i >= 0, at least one positive

Example

m.ask("Leslie growth rate: F = [0, 2], S = [0.5]")

→ lambda1 = 1, r = 0, R0 = 1, stable = [0.666666666667, 0.333333333333]

Leslie matrix population projectionbio_leslie_projection

The age-class numbers and total population after t time steps of an age-structured (Leslie) model.

\mathbf n_{t+1} = L\,\mathbf n_t,\quad L = \begin{pmatrix}F_1 & F_2 & \cdots\\ S_1 & 0 & \\ & \ddots & \end{pmatrix}

Inputs

F fecundity of each age class vector
S survival from each age class to the next (one fewer than F) vector
n0 initial number in each age class vector
t number of time steps integer

Returns

n_t number in each age class after t steps
total total population after t steps

Applies when

0 < S_i <= 1
F_i >= 0, at least one positive
t <= 500

Example

m.ask("Leslie projection: F = [0, 1.5, 1], S = [0.5, 0.8], n0 = [100, 50, 20], t = 1")

→ n_t = [95.0, 50.0, 40.0], total = 185

Population genetics 16

Allele frequency under haploid (genic) selectionbio_haploid_selection

The frequency of allele A after t generations of haploid selection (relative fitness 1 + s against 1).

p' = \frac{p(1+s)}{1+sp}

Inputs

p0 initial frequency of A
s selection coefficient (fitness of A = 1 + s)
t number of generations integer

Returns

p_t frequency of A after t generations

Applies when

0 < p0 < 1
s > -1
discrete generations

Example

m.ask("Haploid selection: p0 = 0.5, s = 1, t = 1")

→ p_t = 0.666667

Decline of a recessive lethal allelebio_recessive_lethal

The frequency of a recessive lethal allele after t generations of selection (aa never reproduce).

q' = \frac{q}{1+q}

Inputs

q0 initial frequency of the lethal allele
t number of generations integer

Returns

q_t frequency after t generations

Applies when

0 < q0 <= 1
random mating, aa fitness 0

Example

m.ask("Recessive lethal allele frequency: q0 = 0.5, t = 1")

→ q_t = 0.333333

Effective population size with unequal sex ratiobio_ne_sex_ratio

The effective population size of a population with N_m breeding males and N_f breeding females.

N_e = \frac{4N_mN_f}{N_m+N_f}

Inputs

N_m number of breeding males integer
N_f number of breeding females integer

Returns

Ne effective population size
Ne_over_N Ne divided by the census N_m + N_f

Applies when

random mating within the breeding population
Wright-Fisher reproduction

Example

m.ask("Effective population size with unequal sex ratio: N_m = 10, N_f = 90")

→ Ne = 36, Ne_over_N = 0.36

Effective size of a fluctuating populationbio_ne_fluctuating_size

The effective population size over generations of changing size (harmonic mean) and the fraction of heterozygosity retained over those generations.

\frac{1}{N_e} = \frac{1}{t}\sum_{i=1}^t \frac{1}{N_i},\quad \frac{H_t}{H_0} = \prod_i \Big(1-\frac{1}{2N_i}\Big)

Inputs

sizes numbers of diploid individuals in successive generations vector

Returns

Ne effective size (harmonic mean of the sizes)
retained heterozygosity retained, product of (1 - 1/(2N_i))

Applies when

whole positive sizes
Wright-Fisher reproduction each generation

Example

m.ask("Effective size of a fluctuating population: sizes = [1000, 10, 1000]")

→ Ne = 29.4118, retained = 0.94905

Equilibrium under heterozygote advantagebio_heterozygote_advantage

The stable equilibrium frequency of allele A when the heterozygote is fittest (overdominance).

\hat p = \frac{w_{Aa} - w_{aa}}{2w_{Aa} - w_{AA} - w_{aa}} = \frac{t}{s+t}

Inputs

w_AA relative fitness of AA
w_Aa relative fitness of Aa
w_aa relative fitness of aa

Returns

p_eq equilibrium frequency of A

Applies when

w_Aa > w_AA and w_Aa > w_aa
fitnesses positive

Example

m.ask("Heterozygote advantage equilibrium: w_AA = 0.8, w_Aa = 1, w_aa = 0.6")

→ p_eq = 0.666667

Expected coalescence timesbio_coalescent_times

The expected time to the most recent common ancestor and the expected total branch length of the genealogy of n gene copies from a diploid population of effective size Ne (in generations).

E[T_{MRCA}] = 4N_e\left(1 - \frac1n\right),\quad E[L] = 4N_e\sum_{i=1}^{n-1}\frac1i

Inputs

n number of sampled gene copies integer
Ne diploid effective population size

Returns

t_mrca expected TMRCA in generations
total_length expected total branch length in generations

Applies when

neutral Kingman coalescent
diploid: a pair coalesces at rate 1/(2Ne) per generation

Example

m.ask("Coalescent times: n = 2, Ne = 1000")

→ t_mrca = 2000, total_length = 4000

Fixation index FST from subpopulation allele frequenciesbio_fst

Mean within-subpopulation heterozygosity HS, total heterozygosity HT and FST = (HT - HS)/HT for a biallelic locus in equally weighted subpopulations.

F_{ST} = \frac{H_T - H_S}{H_T}

Inputs

p frequency of one allele in each subpopulation vector

Returns

HS mean expected heterozygosity within subpopulations
HT expected heterozygosity of the pooled population
FST fixation index (HT - HS)/HT

Applies when

equal subpopulation weights
biallelic locus
0 < mean p < 1

Example

m.ask("FST: p = [0.2, 0.8]")

→ HS = 0.32, HT = 0.5, FST = 0.36

Fixation probability in the Moran processbio_moran_fixation

The probability that a mutant with relative fitness r, starting from i copies in a Moran population of size N, eventually takes over the population.

\rho_i = \frac{1 - r^{-i}}{1 - r^{-N}}

Inputs

N population size integer
i initial number of mutants integer
r relative fitness of the mutant

Returns

rho fixation probability

Applies when

0 < i < N
r > 0

Example

m.ask("Moran fixation probability: N = 10, i = 1, r = 1")

→ rho = 0.1

Hardy-Weinberg chi-square testbio_hardy_weinberg_test

Allele frequencies, Hardy-Weinberg expected genotype counts, the chi-square test of equilibrium (1 degree of freedom), observed and expected heterozygosity and the inbreeding coefficient F.

\chi^2 = \sum \frac{(O-E)^2}{E},\quad E = (Np^2,\ 2Npq,\ Nq^2)

Inputs

n_AA number of AA individuals integer
n_Aa number of Aa individuals integer
n_aa number of aa individuals integer

Returns

p frequency of allele A
q frequency of allele a
exp_AA expected AA count (N p^2)
exp_Aa expected Aa count (2 N p q)
exp_aa expected aa count (N q^2)
chi2 chi-square statistic
p_value p-value (chi-square, 1 degree of freedom)
Ho observed heterozygosity
He expected heterozygosity 2pq
F inbreeding coefficient 1 - Ho/He

Applies when

both alleles present
biallelic locus
no continuity correction

Example

m.ask("Hardy-Weinberg test: n_AA = 298, n_Aa = 489, n_aa = 213")

→ p = 0.5425, q = 0.4575, exp_AA = 294.306, exp_Aa = 496.387, exp_aa = 209.306, chi2 = 0.22149, p_value = 0.637907, Ho = 0.489, He = 0.496387, F = 0.0148825

Hardy-Weinberg exact testbio_hardy_weinberg_exact_test

The exact (Wigginton) test of Hardy-Weinberg equilibrium for a biallelic locus: the two-sided p-value and the probability of the observed number of heterozygotes given the allele counts.

P(n_{AB}\mid n_A) = \frac{N!}{n_{AA}!\,n_{AB}!\,n_{BB}!}\,\frac{2^{n_{AB}}\,n_A!\,n_B!}{(2N)!}

Inputs

n_AA number of AA individuals integer
n_Aa number of Aa individuals integer
n_aa number of aa individuals integer

Returns

p_value exact two-sided p-value (sum of probabilities not above the observed one)
p_observed probability of the observed heterozygote count

Applies when

both alleles present
N <= 1000

Example

m.ask("Hardy-Weinberg exact test: n_AA = 2, n_Aa = 0, n_aa = 2")

→ p_value = 0.0857143, p_observed = 0.0857143

Linkage disequilibrium D, D' and r^2bio_linkage_disequilibrium

The linkage disequilibrium between two biallelic loci from haplotype counts: D, Lewontin's D' and r^2.

D = p_{AB} - p_Ap_B,\quad r^2 = \frac{D^2}{p_Aq_Ap_Bq_B},\quad D' = \frac{D}{D_{\max}}

Inputs

n_AB number of AB haplotypes integer
n_Ab number of Ab haplotypes integer
n_aB number of aB haplotypes integer
n_ab number of ab haplotypes integer

Returns

D disequilibrium D = pAB - pA pB
D_prime Lewontin's D' = D / Dmax (signed)
r2 squared correlation r^2

Applies when

both loci polymorphic
phased haplotype counts

Example

m.ask("Linkage disequilibrium: n_AB = 40, n_Ab = 10, n_aB = 10, n_ab = 40")

→ D = 0.15, D_prime = 0.6, r2 = 0.36

Loss of heterozygosity by genetic driftbio_drift_heterozygosity

The expected heterozygosity after t generations of genetic drift in a Wright-Fisher population of N diploids, starting from i copies of an allele.

H_t = H_0\left(1 - \frac{1}{2N}\right)^t

Inputs

N number of diploid individuals integer
i initial copies of allele A (of 2N) integer
t number of generations integer

Returns

H0 initial heterozygosity 2pq
H_t expected heterozygosity after t generations
retained fraction of heterozygosity retained, (1 - 1/(2N))^t

Applies when

0 < i < 2N
constant N, random mating, no mutation or selection

Example

m.ask("Drift heterozygosity: N = 10, i = 10, t = 1")

→ H0 = 0.5, H_t = 0.475, retained = 0.95

Mutation-selection balancebio_mutation_selection_balance

The equilibrium frequency of a deleterious allele maintained by recurrent mutation against selection (fitness 1, 1 - hs, 1 - s), from the exact recursion.

\hat q \approx \sqrt{\mu/s}\ (h=0),\quad \hat q \approx \frac{\mu}{hs}\ (h>0)

Inputs

mu mutation rate A -> a per generation
s selection coefficient against aa
h dominance coefficient (0 recessive, 0.5 additive)

Returns

q_eq equilibrium frequency of the deleterious allele

Applies when

0 < mu <= 0.01
0.01 <= s <= 1
0 <= h <= 1
selection then mutation, no back mutation

Example

m.ask("Mutation-selection balance: mu = 0.0001, s = 1, h = 0")

→ q_eq = 0.01

Nucleotide diversitybio_nucleotide_diversity

The average number of pairwise differences k and the nucleotide diversity pi (per site) of a set of aligned DNA sequences.

\pi = \frac{1}{L}\binom{n}{2}^{-1}\sum_{i<j} d_{ij}

Inputs

seqs aligned DNA sequences separated by spaces text

Returns

k mean number of pairwise differences
pi nucleotide diversity per site

Applies when

at least 2 aligned DNA sequences of equal length, no gaps

Example

m.ask("Nucleotide diversity: seqs = AAAA AAAT AATT")

→ k = 1.33333, pi = 0.333333

Tajima's Dbio_tajima_d

Tajima's D neutrality statistic of aligned DNA sequences (the scaled difference between pi and Watterson's theta).

D = \frac{\hat\theta_\pi - \hat\theta_W}{\sqrt{e_1S + e_2S(S-1)}}

Inputs

seqs aligned DNA sequences separated by spaces (at least 4) text

Returns

D Tajima's D
k mean number of pairwise differences

Applies when

at least 4 aligned DNA sequences
at least one segregating site

Example

m.ask("Tajima's D: seqs = AAAAAA AAAAAT AAAATT AAATTT")

→ D = 0.167656, k = 1.66667

Watterson's estimator of thetabio_watterson_theta

The number of segregating sites S and Watterson's estimator of the population mutation rate theta, per sequence and per site.

\hat\theta_W = \frac{S}{a_n},\quad a_n = \sum_{i=1}^{n-1}\frac1i

Inputs

seqs aligned DNA sequences separated by spaces text

Returns

S number of segregating sites
theta Watterson's theta per sequence (S / a_n)
theta_site Watterson's theta per site

Applies when

at least 2 aligned DNA sequences of equal length, no gaps

Example

m.ask("Watterson theta: seqs = AAAA AAAT AATT")

→ S = 2, theta = 1.33333, theta_site = 0.333333

Sequences 21

Amino-acid composition of a proteinbio_amino_acid_composition

The percentage of each of the 20 standard amino acids in a protein, its most common residue (alphabetically first among ties) and its length.

\%a = 100\,\frac{n_a}{n}

Inputs

seq protein sequence text

Returns

percent percentage of each amino acid (one-letter code -> %)
most_common most common amino acid (alphabetically first among ties)
length number of residues

Applies when

standard amino acids only

Example

m.ask("Amino acid composition: seq = MAAAK")

→ percent = {'A': 60.0, 'C': 0.0, 'D': 0.0, 'E': 0.0, 'F': 0.0, 'G': 0.0, 'H': 0.0, 'I': 0.0, 'K': 20.0, 'L': 0.0, 'M': 20.0, 'N': 0.0, 'P': 0.0, 'Q': 0.0, 'R': 0.0, 'S': 0.0, 'T': 0.0, 'V': 0.0, 'W': 0.0, 'Y': 0.0}, most_common = A, length = 5

Aromaticity of a proteinbio_aromaticity

The aromaticity (relative frequency of Phe + Trp + Tyr, Lobry 1994) of a protein and its number of aromatic residues.

\mathrm{Ar} = \frac{n_F + n_W + n_Y}{n}

Inputs

seq protein sequence text

Returns

aromaticity fraction of F, W and Y residues
aromatic_count number of F, W and Y residues

Applies when

standard amino acids only

Example

m.ask("Aromaticity: seq = MFWYAK")

→ aromaticity = 0.5, aromatic_count = 3

Edit (Levenshtein) distance between sequencesbio_edit_distance

The minimum number of substitutions, insertions and deletions turning one sequence into the other (unit costs; an unaligned comparison, unlike the p-distance).

D_{i,j} = \min(D_{i-1,j}+1,\ D_{i,j-1}+1,\ D_{i-1,j-1}+[x_i\ne y_j])

Inputs

seq1 first sequence text
seq2 second sequence text

Returns

distance edit distance (unit costs)

Applies when

letters only

Example

m.ask("Edit distance: seq1 = KITTEN, seq2 = SITTING")

→ distance = 3

GC content of a nucleic acidbio_gc_content

The fraction and number of G and C bases in a DNA or RNA sequence.

\mathrm{GC} = \frac{n_G + n_C}{n}

Inputs

seq DNA or RNA sequence text

Returns

gc_fraction fraction of G + C
gc_count number of G and C bases

Applies when

unambiguous DNA or RNA letters

Example

m.ask("GC content: seq = ATGCGCGA")

→ gc_fraction = 0.625, gc_count = 5

Grand average of hydropathy (GRAVY)bio_gravy

The Kyte-Doolittle grand average of hydropathy of a protein (positive = hydrophobic).

\mathrm{GRAVY} = \frac{1}{n}\sum_i h(a_i)

Inputs

seq protein sequence text

Returns

gravy GRAVY score

Applies when

standard amino acids only

Example

m.ask("GRAVY: seq = MKTLLIL")

→ gravy = 1.88571

Isoelectric point of a proteinbio_isoelectric_point

The isoelectric point (pH of zero net charge) of a protein, with its net charge at pH 7 (Bjellqvist pK set).

Z(\mathrm{pI}) = 0

Inputs

seq protein sequence text

Returns

pI isoelectric point
charge_at_7 net charge at pH 7

Applies when

standard amino acids only

Example

m.ask("Isoelectric point: seq = INGAR")

→ pI = 9.75002, charge_at_7 = 0.760092

k-mer counts of a sequencebio_kmer_counts

Overlapping k-mer counts: the number of k-mers, of distinct k-mers, the highest count and the most frequent k-mer (alphabetically first among ties).

N_k = n - k + 1

Inputs

seq DNA or RNA sequence text
k k-mer length integer

Returns

total number of overlapping k-mers (n - k + 1)
distinct number of distinct k-mers
max_count count of the most frequent k-mer
most_frequent most frequent k-mer (alphabetically first among ties)

Applies when

1 <= k <= sequence length

Example

m.ask("k-mer counts: seq = ATATAT, k = 2")

→ total = 5, distinct = 2, max_count = 3, most_frequent = AT

Longest open reading framebio_longest_orf

The longest open reading frame (ATG to the first in-frame stop, stop included) over all six frames of a DNA sequence, its protein length and the number of ORFs.

\ell = 3(j - i + 1)\ \text{for ATG at codon } i,\ \text{first stop at codon } j

Inputs

seq DNA sequence 5'->3' text

Returns

length_nt length of the longest ORF in nucleotides, stop codon included
protein_length amino acids encoded by the longest ORF (stop excluded)
orf_count number of ORFs (one per stop codon reached from an upstream in-frame ATG)

Applies when

DNA letters only
standard genetic code, start ATG, stops TAA TAG TGA

Example

m.ask("Longest ORF: seq = CCATGAAATTTGGGTAACC")

→ length_nt = 15, protein_length = 4, orf_count = 1

Melting temperature by nearest-neighbour thermodynamicsbio_tm_nearest_neighbor

The melting temperature of a DNA duplex (perfect complement) from nearest-neighbour thermodynamics (Allawi-SantaLucia 1997) with the SantaLucia 1998 sodium correction.

T_m = \frac{1000\,\Delta H}{\Delta S + 0.368(N-1)\ln[\mathrm{Na^+}] + R\ln k} - 273.15

Inputs

seq DNA sequence 5'->3' text
Na sodium concentration in mM
dnac1 concentration of the more concentrated strand in nM
dnac2 concentration of the less concentrated strand in nM

Returns

tm melting temperature in degC

Applies when

unambiguous DNA, length >= 2
Na > 0
dnac1 > dnac2/2

Example

m.ask("Nearest-neighbour melting temperature: seq = CGTTCCAAAGATGTGGGCATGAGCTTAC, Na = 50, dnac1 = 25, dnac2 = 25")

→ tm = 60.3209

Melting temperature by the Wallace rulebio_tm_wallace

The approximate melting temperature of a short DNA primer (14-20 nt) by the Wallace rule.

T_m = 2(n_A+n_T) + 4(n_G+n_C)\ ^\circ\mathrm{C}

Inputs

seq DNA primer sequence text

Returns

tm melting temperature in degC

Applies when

unambiguous DNA letters

Example

m.ask("Wallace melting temperature: seq = ACGTTGCAATGCCGTA")

→ tm = 48

Molecular mass of a proteinbio_protein_mass

The average molecular mass of a protein from its amino-acid sequence.

M = \sum_i m(a_i) - (n-1)\,m_{\mathrm{H_2O}}

Inputs

seq protein sequence (20 standard one-letter codes) text

Returns

mass molecular mass in g/mol (Da)

Applies when

standard amino acids only

Example

m.ask("Protein mass: seq = MAEGEITTFTALTEKFNLPPGNYKKPKLLYCSNGGHFLRILPDGTVDGTRDRSDQHIQLQLSAESVGEVYIKSTETGQYLAMDTSGLLYGSQTPSEECLFLERLEENHYNTYTSKKHAEKNWFVGLKKNGSCKRGPRTHYGQKAILFLPLPV")

→ mass = 17103.2

Molecular mass of DNA or RNAbio_nucleic_acid_mass

The average molecular mass of a single- or double-stranded linear DNA or RNA with a 5' phosphate.

M = \sum_i m(x_i) - (n-1)\,m_{\mathrm{H_2O}}

Inputs

seq DNA or RNA sequence text
molecule molecule type choice
strands single- or double-stranded choice

Returns

mass molecular mass in g/mol (Da)

Applies when

the sequence letters match the stated molecule

Example

m.ask("Nucleic acid mass: seq = AGC, molecule = DNA, strands = single")

→ mass = 949.61

Net charge of a protein at a pHbio_protein_net_charge

The net charge of a protein at a stated pH from its ionisable groups (Bjellqvist pK set).

Z = \sum_{\text{basic}} \frac{n_i}{1+10^{\,pH-pK_i}} - \sum_{\text{acidic}} \frac{n_j}{1+10^{\,pK_j-pH}}

Inputs

seq protein sequence text
pH pH of the solution (0 to 14)

Returns

charge net charge (elementary charges)

Applies when

standard amino acids only
0 <= pH <= 14

Example

m.ask("Protein net charge: seq = INGAR, pH = 7")

→ charge = 0.760092

p-distance (Hamming) between aligned sequencesbio_hamming_distance

Mismatches, p-distance and identity of two aligned sequences of equal length (no indels): the proportion of sites that differ.

p = \frac{1}{n}\sum_i [x_i \ne y_i]

Inputs

seq1 first aligned sequence text
seq2 second aligned sequence (same length) text

Returns

mismatches number of differing sites
p_distance proportion of differing sites
identity proportion of identical sites

Applies when

equal lengths (aligned, no gaps)

Example

m.ask("p-distance: seq1 = GAGCCTACTAACGGGAT, seq2 = CATCGTAATGACGGCCT")

→ mismatches = 7, p_distance = 0.411765, identity = 0.588235

Protein extinction coefficient at 280 nmbio_protein_extinction

The molar extinction coefficient at 280 nm (cysteines reduced, and all cysteines as cystines) and the absorbance of a 1 g/L solution.

\varepsilon_{280} = 5500\,n_W + 1490\,n_Y + 125\,n_{\text{cystine}}

Inputs

seq protein sequence text

Returns

eps_reduced extinction coefficient, cysteines reduced (M^-1 cm^-1)
eps_cystines extinction coefficient, cysteines paired as cystines (M^-1 cm^-1)
abs_reduced absorbance of 1 g/L, 1 cm, cysteines reduced
abs_cystines absorbance of 1 g/L, 1 cm, cystines

Applies when

standard amino acids only

Example

m.ask("Protein extinction coefficient: seq = MAEGEITTFTALTEKFNLPPGNYKKPKLLYCSNGGHFLRILPDGTVDGTRDRSDQHIQLQLSAESVGEVYIKSTETGQYLAMDTSGLLYGSQTPSEECLFLERLEENHYNTYTSKKHAEKNWFVGLKKNGSCKRGPRTHYGQKAILFLPLPV")

→ eps_reduced = 17420, eps_cystines = 17545, abs_reduced = 1.01853, abs_cystines = 1.02583

Protein instability indexbio_instability_index

The Guruprasad instability index of a protein from its dipeptides (above 40 suggests an unstable protein in vivo).

II = \frac{10}{L}\sum_{i=1}^{L-1} \mathrm{DIWV}(a_i, a_{i+1})

Inputs

seq protein sequence (at least 2 residues) text

Returns

index instability index

Applies when

standard amino acids, length >= 2

Example

m.ask("Instability index: seq = MAEGEITTFTALTEKFNLPPGNYKKPKLLYCSNGGHFLRILPDGTVDGTRDRSDQHIQLQLSAESVGEVYIKSTETGQYLAMDTSGLLYGSQTPSEECLFLERLEENHYNTYTSKKHAEKNWFVGLKKNGSCKRGPRTHYGQKAILFLPLPV")

→ index = 41.9803

Restriction sites and fragmentsbio_restriction_sites

Where a restriction enzyme cuts a linear DNA sequence: the number of sites, the cut positions and the fragment lengths.

\text{cut at } i + c \text{ for each site start } i

Inputs

seq linear DNA sequence 5'->3' text
enzyme restriction enzyme (name) choice

Returns

n_sites number of cuts
positions cut positions (1-based first base after each cut, top strand)
fragments fragment lengths in bp, 5'->3' order

Applies when

linear DNA
a listed enzyme

Example

m.ask("Restriction sites: seq = AAGAATTCTTGAATTCGGATCC, enzyme = EcoRI")

→ n_sites = 2, positions = [4, 12], fragments = [3, 8, 11]

Reverse complement of a DNA sequencebio_reverse_complement

The reverse complement of a DNA strand (the partner strand read 5'->3').

5'\text{-}x_1\dots x_n\text{-}3' \mapsto 5'\text{-}\bar x_n\dots \bar x_1\text{-}3'

Inputs

seq DNA sequence 5'->3' (A, C, G, T) text

Returns

revcomp reverse complement 5'->3'

Applies when

DNA letters A, C, G, T only

Example

m.ask("Reverse complement: seq = ATGCCGTA")

→ revcomp = TACGGCAT

Secondary-structure propensity fractionsbio_secondary_structure_fraction

The fractions of residues that tend to be in helices (E M A L K), turns (N P G S D) and sheets (V I Y F W L T), as ProtParam defines them (L counts for helix and sheet).

f_{\text{helix}} = \frac{n_E+n_M+n_A+n_L+n_K}{n}

Inputs

seq protein sequence text

Returns

helix fraction of helix-favouring residues
turn fraction of turn-favouring residues
sheet fraction of sheet-favouring residues

Applies when

standard amino acids only

Example

m.ask("Secondary structure fraction: seq = EMALKNPGSD")

→ helix = 0.5, turn = 0.5, sheet = 0.1

Transcription of DNA into mRNAbio_transcription

The mRNA transcribed from a DNA strand given 5'->3', stated as the coding (sense) or template (antisense) strand.

\text{mRNA} = \text{coding strand with } T\to U = \overline{\text{template}}^{\,R}

Inputs

seq DNA strand 5'->3' (A, C, G, T) text
strand which strand is given: coding (sense) or template (antisense) choice

Returns

mrna mRNA 5'->3'

Applies when

DNA letters only
the strand is stated

Example

m.ask("Transcription: seq = ATGGCTTAA, strand = coding")

→ mrna = AUGGCUUAA

Translation with the standard genetic codebio_translation

The protein encoded by a DNA or mRNA sequence read in frame 1 with the standard genetic code; stop codons shown as '*' or the protein truncated at the first stop.

\text{codon } x_{3i+1}x_{3i+2}x_{3i+3} \mapsto \text{amino acid}_i

Inputs

seq DNA or mRNA sequence 5'->3', length a multiple of 3 text
stops show stop codons as '*' or truncate at the first stop choice

Returns

protein amino-acid sequence (one-letter codes)
length number of residues shown

Applies when

length a multiple of 3
DNA or RNA letters, not a mixture

Example

m.ask("Translation: seq = ATGGCCATTGTAATGGGCCGCTGA, stops = show")

→ protein = MAIVMGR*, length = 8

Systems biology 1

Flux balance analysis (optimal objective)bio_flux_balance_analysis

The maximum of a linear objective (e.g. a biomass or product flux) over all steady-state flux distributions of a metabolic network within flux bounds: flux balance analysis.

\max_v\ c^\top v \ \text{ s.t. } Sv = 0,\ \ l \le v \le u

Inputs

S stoichiometric matrix (metabolites x reactions) matrix
lb lower bound of each reaction flux vector
ub upper bound of each reaction flux vector
c objective coefficient of each reaction vector

Returns

objective optimal objective value

Applies when

steady state S v = 0
finite bounds lb <= ub
a feasible flux distribution exists

Example

m.ask("Flux balance analysis: S = [[1, -1, 0, 0], [0, 1, -1, -1]], lb = [0, 0, 0, 0], ub = [10, 8, 100, 3], c = [0, 0, 1, 0]")

→ objective = 8

Versions & files

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model.safetensors	18.9 MB
dist/computational_biology_ai999x-1.0.0-py3-none-any.whl	14.8 MB
onnx/search/encoder.int8.onnx	4.9 MB
docs/computational_biology_ai999x_capabilities.pdf	2.8 MB
onnx/search/index.json	403.3 KB
docs/computational_biology_ai999x_capabilities.html	266.4 KB
docs/computational_biology_ai999x_capabilities.md	202.4 KB
onnx/search/tokenizer.json	176.4 KB
README.md	20.8 KB
LICENSE	11.4 KB
evaluation/release_gate.md	2.3 KB
evaluation/KNOWLEDGE_CARD.md	1.4 KB

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Questions

Frequently asked

What does computational-biology-ai999x do?

Computational Biology answers questions in computational biology: DNA and protein sequences, alignment, molecular weights, genetics and population genetics, phylogenetics, enzyme kinetics, metabolic and population models, and ecology. Each answer gives the value with its unit, the steps and the formula in LaTeX.

How do I install computational-biology-ai999x?

With pip: pip install computational-biology-ai999x. The same files are on Hugging Face (hf download AI999X/computational-biology-ai999x --local-dir computational-biology-ai999x) and GitHub.

Does it run offline?

Yes. It is a Python package that runs on an ordinary CPU without a network connection (Python >=3.10); no GPU and no API key are needed.

How many capabilities does computational-biology-ai999x have?

83 capabilities in 10 topics, including Sequences, Population genetics, Ecology, Enzyme kinetics, Alignment.

Which field and tasks does it cover?

Field: Life Sciences. Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction.

What does an answer look like?

Asked "Reverse complement: seq = ATGCCGTA", it answers: revcomp = TACGGCAT.

What happens when a question cannot be answered?

It says so and names what is missing or out of range, instead of guessing a number.

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