Parameterization of the Model

iTRAILS leverages the following population parameters in order to infer the underlying gene tree topologies and coalescent event timings (see Figure iTRAILS_params).

Population Size Parameters

  • N_AB: Effective size of the ancestral population between the two most closely related species.

  • N_ABC: Effective size of the ancestral population between the three species.

Time Parameters (generations)

  • T1: Assuming an ultrametric model, time from sample to coalescence of species A and B.

  • T2: Time between the coalescence of species A and B and the coalescence of AB ancestral with species C.

  • T3: Time between coalescence of species ABC and coalescence of ABC ancestral with outgroup species.

  • T_A: Time from sample of species A to coalescence with species B.

  • T_B: Time from sample of species B to coalescence with species A.

  • T_C: Time from sample of species C to coalescence with AB ancestral.

  • T_Upper: Time from the start of the last discretized time interval in ABC ancestral to coalescence with the outgroup.

  • T_Out: Time from sample of the outgroup to coalescence with ancestral ABC.

Other Parameters

  • Recombination rate (ρ): Number of recombinations per site per generation.

  • Mutation rate (μ): Number of mutations per site per generation.

Dependencies and Restrictions

  • Recombination rate (ρ) must always be defined either as a fixed parameter or as a parameter to optimize.

  • Mutation rate (μ) must always be defined as a fixed parameter.

  • T_Upper and T3: At least one of them must be defined (fixed or optimized). If only one is defined, the other is automatically calculated.

  • T_Out: Can be defined or omitted; if defined, it must be fixed. If omitted, it will be automatically calculated.

  • T_A, T_B, T_C, and T1 should be defined (fixed or optimized) in the following combinations. The non-defined parameters will take values as indicated in the table below.

Relationship Between T_A, T_B, T_C, and T1

Specified and Used Time Parameters

Specified Parameters

Used T_A

Used T_B

Used T_C

T_A / T_B / T_C

T_A

T_B

T_C

T1 / T_A

T_A

T1

T1 + T2

T1 / T_B

T1

T_B

T1 + T2

T1 / T_C

T1

T1

T_C

T_A / T_B

T_A

T_B

(T_A + T_B) / 2 + T2

T_A / T_C

T_A

(T_A + T_C - T2) / 2

T_C

T_B / T_C

(T_B + T_C - T2) / 2

T_B

T_C

T1

T1

T1

T1 + T2