Table 1 Repeat instability rate parameterizations and Bayesian inference results

From: Inherent instability of simple DNA repeats shapes an evolutionarily stable distribution of repeat lengths

Model name

Parameters

Functional form

Prior

Bayes factor (ratio to null)

Max posterior

[mean posterior]

Symmetric power law†

(L < 9 from empirical rates)

2;

\(\theta=(c,\tau)\)

\(\epsilon \left(L > 8\right)=\kappa \left(L > 8\right)=c{\left(\frac{L}{9}\right)}^{\tau }\)

Uniform

5.1e-18 (1.0)

\(\theta=\)(4.3e-8, 0.6)

[(4.3e-8, 0.67)]

Permissive

2.8e-21 (1.0)

\(\theta=\)(4.3e-8,1.2)

[(4.1e-8, 1.3)]

Restrictive

7.6e-31 (1.0)

\(\theta=\)(2.7e-8, 3.0)

[(3.4e-8, 2.8)]

Decoupled power laws

(L < 9 empirical)

4;

\(\theta=({c}_{\epsilon },{c}_{\kappa },{\tau }_{\epsilon },\,{\tau }_{\kappa })\)

\(\epsilon \left(L > 8\right)={c}_{\epsilon }{\left(\frac{L}{9}\right)}^{{\tau }_{\epsilon }}\)

\(\kappa \left(L > 8\right)={c}_{\kappa }{\left(\frac{L}{9}\right)}^{{\tau }_{\kappa }}\)

Uniform

2.0e-5 (3.9e12)

\(\theta=\)(2.7e-8, 1.7e-8, 1.6, 2.0)

[(2.9e-8, 1.8e-8, 1.9, 2.3)]

Permissive

1.9e-5 (6.9e15)

\(\theta=\)(2.7e-8, 1.7e-8, 3.0, 3.5)

[(3.9e-8, 2.5e-8, 2.9, 3.3)]

Restrictive

2.2e-5 (2.9e25)

\(\theta=\) (4.3e-8, 2.7e-8, 3.1, 3.6)

[(4.7e-8, 2.9e-8, 3.1, 3.6)]

Power laws with independent constants

(L < 9 empirical)

3;

\(\theta=({c}_{\epsilon },{c}_{\kappa },\tau )\)

\(\epsilon \left(L > 8\right)={c}_{\epsilon }{\left(\frac{L}{9}\right)}^{\tau }\)

\(\kappa \left(L > 8\right)={c}_{\kappa }{\left(\frac{L}{9}\right)}^{\tau }\)

Uniform

1.3e-5 (2.6e12)

\(\theta=\)(1.7e-8, 1.1e-8, 0.9)

[(2.0e-8, 1.2e-8, 0.6)]

Permissive

9.9e-14 (3.5e7)

\(\theta=\)(1.7e-8, 1.1e-8, 1.0)

[(1.7e-8, 1.1e-8, 1.0)]

Restrictive

2.8e-31 (3.7e-1)

\(\theta=\) (2.7e-8, 2.7e-8, 3.0)

[(3.4e-8, 3.4e-8, 2.8)]

Power laws with independent exponents

(L < 9 empirical)

3;

\(\theta=(c,{\tau }_{\epsilon },\,{\tau }_{\kappa })\)

\(\epsilon \left(L > 8\right)=c{\left(\frac{L}{9}\right)}^{{\tau }_{\epsilon }}\)

\(\kappa \left(L > 8\right)=c{\left(\frac{L}{9}\right)}^{{\tau }_{\kappa }}\)

Uniform

9.7e-7 (1.9e11)

\(\theta=\)(2.7e-8, 0.6, 0.1)

[(2.7e-8, 2.3, 2.1)]

Permissive

2.3e-6 (8.2e14)

\(\theta=\)(2.7e-8, 3.4, 3.3)

[(2.8e-8, 3.3, 3.2)]

Restrictive

2.0e-6 (2.6e24)

\(\theta=\) (2.7e-8, 3.6, 3.5)

[(2.9e-8, 3.5, 3.4)]

Multiplier-coupled power laws

(L < 9 empirical)

3;

\({{{\boldsymbol{\theta }}}}=({{{\boldsymbol{m}}}},{{{{\boldsymbol{\tau }}}}}_{{{{\boldsymbol{\epsilon }}}}},\,{{{{\boldsymbol{\tau }}}}}_{{{{\boldsymbol{\kappa }}}}})\)

\({{{\boldsymbol{\epsilon }}}}\left({{{\boldsymbol{L}}}}{{{\boldsymbol{ > }}}}{{{\bf{8}}}}\right)={{{\boldsymbol{\epsilon }}}}\left({{{\bf{8}}}}\right){{{\boldsymbol{\times }}}}{{{\boldsymbol{m}}}}{\left(\frac{{{{\boldsymbol{L}}}}}{{{{\bf{9}}}}}\right)}^{{{{{\boldsymbol{\tau }}}}}_{{{{\boldsymbol{\epsilon }}}}}}\)

\({{{\boldsymbol{\kappa }}}}\left({{{\boldsymbol{L}}}}{{{\boldsymbol{ > }}}}{{{\bf{8}}}}\right)={{{\boldsymbol{\kappa }}}}\left({{{\bf{8}}}}\right){{{\boldsymbol{\times }}}}{{{\boldsymbol{m}}}}{\left(\frac{{{{\boldsymbol{L}}}}}{{{{\bf{9}}}}}\right)}^{{{{{\boldsymbol{\tau }}}}}_{{{{\boldsymbol{\kappa }}}}}}\)

(\({{{\boldsymbol{\epsilon }}}}\left({{{\bf{8}}}}\right),\,{{{\boldsymbol{\kappa }}}}\left({{{\bf{8}}}}\right)\) from empirical estimates)

Uniform

1.8e-4 (3.5e13)

\({{{\boldsymbol{\theta }}}}=\)(2.5, 1.6, 2.0)

[(2.7, 1.9, 2.3)]

Permissive

1.4e-4 (4.8e16)

\({{{\boldsymbol{\theta }}}}=\)(4.0, 2.8, 3.3)

[(3.6, 2.9, 3.3)]

Restrictive

1.3e-4 (1.7e26)

\({{{\boldsymbol{\theta }}}}=\) (4.0, 3.1, 3.6)

[(4.3, 3.1, 3.6)]

Symmetric logarithmic power††

(L < 9 from empirical rates)

2;

\(\theta=(c,\tau )\)

\(\epsilon \left(L > 8\right)=\kappa \left(L > 8\right)\,=c{\left(\frac{\log (L-7)}{\log 2}\right)}^{\tau }\)

Uniform

8.4e-18 (1.0)

\(\theta=\)(2.7e-8, 0.8)

[(3.2e-8, 0.7)]

Permissive

3.4e-19 (1.0)

\(\theta=\)(2.7e-8, 0.9)

[(3.4e-8, 0.7)]

Restrictive

8.3e-29 (1.0)

\(\theta=\)(4.3e-8, 1.0)

[(4.2e-8, 1.0)]

Decoupled logarithmic power

(L < 9 empirical)

4;

\(\theta=({c}_{\epsilon },{c}_{\kappa },{\tau }_{\epsilon },\,{\tau }_{\kappa })\)

\(\epsilon \left(L > 8\right)\,={c}_{\epsilon }{\left(\frac{\log \left(L-7\right)}{\log 2}\right)}^{{\tau }_{\epsilon }}\)

\(\kappa \left(L > 8\right)\,={c}_{\kappa }{\left(\frac{\log \left(L-7\right)}{\log 2}\right)}^{{\tau }_{\kappa }}\)

Uniform

1.8e-4 (2.1e13)

\(\theta=\)(2.7e-8, 1.1e-8, 2.1, 2.8)

[(3.4e-8, 1.4e-8, 1.7, 2.4)]

Permissive

5.1e-4 (1.5e15)

\(\theta=\)(4.3e-8, 1.7e-8, 2.0, 2.7)

[(3.8e-8, 1.4e-8, 2.1, 2.9)]

Restrictive

1.8e-3 (2.2e25)

\(\theta=\)(4.3e-8, 1.7e-8, 2.0, 2.7)

[(3.8e-8, 1.3e-8, 2.2, 3.1)]

Multiplier-coupled logarithmic power

(L < 9 empirical)

3;

\(\theta=(m,{\tau }_{\epsilon },\,{\tau }_{\kappa })\)

\(\epsilon \left(L > 8\right)\,=\epsilon \left(8\right){m}{\left(\frac{\log \left(L-7\right)}{\log 2}\right)}^{{\tau }_{\epsilon }}\)

\(\kappa \left(L > 8\right)\,=\kappa \left(8\right){m}{\left(\frac{\log \left(L-7\right)}{\log 2}\right)}^{{\tau }_{\kappa }}\)

(\(\epsilon \left(8\right),\,\kappa \left(8\right)\) from empirical estimates)

Uniform

3.9e-4 (4.6e13)

\(\theta=\)(2.5, 1.3, 0.9)

[(2.8, 1.0, 1.2)]

Permissive

2.7e-4 (7.7e14)

\(\theta=\)(4.0, 1.4, 1.7)

[(3.5, 1.5, 1.8)]

Restrictive

1.7e-4 (2.0e24)

\(\theta=\)(4.0, 1.8, 2.1)

[(4.0, 1.9, 2.2)]

Pure power law

(parameterized at all lengths)

4;

\(\theta=({\lambda }_{\epsilon },{\lambda }_{\kappa },{\tau }_{\epsilon },\,{\tau }_{\kappa })\)

\(\epsilon \left(L\right)=\mu {\left(\frac{L}{{\lambda }_{\epsilon }}\right)}^{{\tau }_{\epsilon }}\)

\(\kappa \left(L\right)=\nu {\left(\frac{L}{{\lambda }_{\kappa }}\right)}^{{\tau }_{\kappa }}\)

(empirically estimated sub. rates μ, ν)

Uniform

3.1e-17

\(\theta=\)(9, 13, 3.6, 4.0)

[(9.1, 12.9, 3.7, 4.0)]

Restrictive

8.7e-17

\(\theta=\)(9, 12, 3.8, 4.0)

[(9.0, 12.5, 3.7, 4.0)]

  1. null model for †power-law and ††logarithmic power models; bold: largest Bayes factor amongst power-law models
  2. Parametric models of instability rates and summary of Bayesian inference results. For each parameterization used in our analyses, this table specifies the model name (as referred to in the text), the tract lengths described by the parameterization, the inference parameters, the functional forms for length-dependent expansion and contraction rates, and a summary of inference results. For each model, the following quantities are given for each prior: Bayes factor (and Bayes factor ratio to null model within the same nesting, denoted by symbols), parameter combination with maximum posterior probability, and mean posterior parameter combination. The primary model considered is shown in bold text. Further details on prior construction, calculation of Bayes factors (and model comparison), and expectation used to compute mean posterior parameters are provided in the “Methods” section.
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