## Highlights

- •MPSproto requires calibration of stutters and noise.
- •MPSproto increases the sensitivity of MPS-STR mixture analyses.
- •MPSproto is a good alternative to EuroForMix.
- •The gamma model in MPSproto performed best overall.

## Abstract

## Keywords

## 1. Introduction

- Barrio P.A.
- et al.

*Forensic Sci. Int. Genet.*2020; 49https://doi.org/10.1016/j.fsigen.2020.102391

- Benschop C.C.G.
- et al.

*Forensic Sci. Int. Genet.*2021; 52https://doi.org/10.1016/j.fsigen.2021.102489

- Li R.
- et al.

*Forensic Sci. Int. Genet.*2020; 45https://doi.org/10.1016/j.fsigen.2019.102225

- Li R.
- et al.

*Forensic Sci. Int. Genet.*2020; 45https://doi.org/10.1016/j.fsigen.2019.102225

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

- Li R.
- et al.

*Forensic Sci. Int. Genet.*2020; 45https://doi.org/10.1016/j.fsigen.2019.102225

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

## 2. Materials and methods

### 2.1 Data

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

### 2.2 The MPSproto model

#### 2.2.1 Models for sequence reads

for the GA model, and for the NB model we have:

*K*individuals (

*K*-person mixture), where each individual contributes with mixture proportion parameter ${\pi}_{k}\in \left[\mathrm{0,1}\right]$. ${n}_{m,a,k}$ is the allele contribution (0, 1 or 2) decided by the assumed genotypes vector ${\mathit{g}}_{m}=\left({g}_{1},{g}_{2},\dots ,{g}_{K}\right)$: ${n}_{m,a,k}=I\left({g}_{k,1}=a\right)+I\left({g}_{k,2}=a\right)$, where ${g}_{k}={g}_{k,1}/{g}_{k,2}$ and $I$ is the indicator function. The marker efficiency parameters $\mathit{A}={(A}_{1},\dots ,{A}_{M})$ are restricted such that ${\frac{1}{M}\sum A}_{m}$= 1. Formulae for the probability density and cumulative functions $f$ and $F$ of the read depths, and the derivation of the size argument are included in the Appendix B, Appendix B.

#### 2.2.2 Stutter model

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

where $*$ indicates the ${\mathrm{\psi}}_{m,a}$ identity without assuming any stutter model (Eq. 1 for the GA model or Eq. 2 for the NB model). The first term is the “donates” part whereas the second term is the “receives” part. When several stutter types ($t=1,\dots $) are defined, the formula extends to:

*.*The formula above hence gives:

where ${{\beta}^{m,t}=(\beta}_{0}^{m,t},{\beta}_{1}^{m,t},{\beta}_{2}^{m,t})$ are model coefficients of the regression and ${x}_{1,a}^{m,t}$ is the BLMM for stutter $a$ and ${x}_{2,a}^{m,t}$ is the BLMM for a different motif (also for stutter $a$). The latter is used for n0 stutters where two parts of the sequence stutter simultaneously, but there is no net change of the size of the stutter product compared to its parent allele.

#### 2.2.3 Noise model

- (A)The number of (unique) noise sequences, $k$, follows a geometric distribution with parameter $p$:$\mathit{Pr}(k\mathrm{|}p)=p{\left(1-p\right)}^{k}$
- (B)The read depth of a noise sequence, $y$, is proportional to a Type I Pareto distribution with parameter $\lambda $ [[22]]:$d\left(y|,\lambda \right)=c*\lambda {T}_{m}^{\lambda}{y}^{-(\lambda +1)}$

#### 2.2.4 Dropout and degradation model

#### 2.2.5 Implementation

### 2.3 Calibration of MPSproto

- 1)Estimation of marker amplification efficiencies
- 2)Identification of stutter types and estimation of regression coefficients
- 3)Estimation of the noise model

#### 2.3.1 Estimating marker amplification efficiency

**,**the read depth sum per marker per sample is used. For each calibration sample $s=1,\dots ,S$, the sum of reads per marker is calculated as ${y}_{\mathrm{s},\mathrm{m}}={\mathrm{\Sigma}}_{\mathrm{a}}{y}_{\mathrm{s},\mathrm{m},\mathrm{a}}$. If ${y}_{s,m}\ge {T}_{m}$ (no marker dropout), we assume that the sum is gamma distributed (for the GA model), with $2{{A}_{m}\omega}_{s}^{-2}$ as the shape argument and ${\mu}_{s}{\omega}_{s}^{2}$ as the scale argument. On the other hand, if ${y}_{s,m}<{T}_{m}$ (marker dropout is present), then the dropout probability is calculated as

where ${F}_{s,m}$is the cumulative density function of the gamma distribution (applied with same shape and scale argument, see Appendix). The estimated marker amplification efficiency parameters are obtained by maximizing the likelihood

#### 2.3.2 Stutter calibration

- •BW1: Backward stutter (
*n-1*) where the longest repeat is stuttering. - •FW1: Forward stutter (
*n+1*) where the longest repeat is stuttering. - •DBW1: Double backward stutter (
*n-2*) where the longest repeat is stuttering. - •BW2: Backward stutter (
*n-1*) where a repeat other than the longest repeat is stuttering. - •FWBW: Longest repeat produces forward stutter,
**and**a repeat other than the longest repeat produces backward stutter (*n0*) - •BWFW: Longest repeat produces backward stutter,
**and**a repeat other than the longest repeat produces forward stutter (*n0*)

*getStutteredSequence*is applied on all donor alleles and compared with artefacts. Stutter artefacts are not considered in the calibration if multiple parental allele candidates are observed.

^{ 1}

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

#### 2.3.3 Estimating noise model

- (A)Let the number of noise sequences per sample be given as ${x}_{1},{x}_{2},\dots ,{x}_{S}$. An additional observation is always appended as ${x}_{S+1}=1+\mathrm{max}\left({x}_{1},\dots ,{x}_{S}\right)$ to make the model more robust for new observations. The $p$ parameter is then estimated as $\stackrel{\u02c6}{p}={\left(\stackrel{\u0305}{x}+1\right)}^{-1}$, where $\stackrel{\u0305}{x}$ is the mean of the observations.
- (B)Let the read depth of the noise sequences be given vectorized as ${y}_{1},{y}_{2},{\dots ,y}_{n}$. An additional observation is always appended as ${y}_{n+1}={T}_{m}$ to make the model more robust for new observations. If $n=0$ (no observations) two additional observations, ${y}_{n+2}={T}_{m}$ and$\phantom{\rule{1em}{0ex}}{y}_{n+3}={T}_{m}+1$, are appended to enable the model to be fitted. The $\lambda $ parameter is then estimated using the maximum likelihood estimate, obtained as $\stackrel{\u02c6}{\lambda}$.

#### 2.3.4 Implementation and functionalities

*calibrateModel*in MPSproto. This eases the calibration steps for users. Additionally, a tutorial is available at https://github.com/oyvble/MPSproto/blob/master/doc/MPSproto_tutorial.html.

### 2.4 Inference of MPS evidence and LR calculations

#### 2.4.1 Definition of the likelihood function and the calculation of LR

**,**whereas the stutter model parameters, marker amplification efficiency and noise model are based on the calibration. As with EuroForMix, the LR calculation is based on optimizing the likelihood function over the model parameters for a specific proposition $H$:

where for a given marker $m$, ${E}_{m}$ is the evidence information (sequences and read depths), and ${\mathit{g}}_{\mathit{m}}=({g}_{1},\dots ,{g}_{K})$ is the joint genotype combination of $K$ contributors, traversed over the joint genotype outcome ${G}_{m}^{H}$ defined from proposition $H$. More specifically:

where $k$ is the assumed number of noise sequences for a given genotype combination (not derived from a contributor or a stutter). If a sequence $a$ is assumed to be a noise sequence, then $f\left({y}_{m,a}|,\theta ,{\mathit{g}}_{\mathit{m}}\right)=d({y}_{m,a}|{\stackrel{\u02c6}{\lambda}}_{m})$

**.**The calculation of the genotype probabilities, $\mathit{Pr}({\mathit{g}}_{\mathit{m}}\mathit{|H})$, includes the ${F}_{\mathit{st}}$coancestry coefficient to take sub-population structure into account, with formulae described in [

#### 2.4.2 Mixture evaluation study

- -${H}_{p}$: “POI + ($K-1$) unknown unrelated are contributors”
- -${H}_{d}$: “$K$ unknown unrelated are contributors”

- -${H}_{p}$: “POI + Ref1 + 2 unknown unrelated are contributors”
- -${H}_{d}$: “Ref1 + 3 unknown unrelated are contributors”

- 1.MPSproto applied with gamma model (GA)
- 2.MPSproto applied with negative binomial model (NB)
- 3.EuroForMix (EFM)

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

#### 2.4.3 Model comparisons

*delong*).

*validMLE*(MPSproto) or

*validMLEmodels*(EuroForMix). A small p-value (e.g. <0.01) from the test indicates a non-adequate model for the read depth.

## 3. Results

### 3.1 Calibration of MPSproto for the mixture evaluation study

Marker | A | p | lambda | BW1 | FW1 | DBW1 | BW2 | FWBW | BWFW |
---|---|---|---|---|---|---|---|---|---|

CSF1PO | 0.324 | 0.989 | 15.6 | -4.44/0.133 | NA | NA | NA | NA | NA |

D10S1248 | 0.629 | 0.91 | 3.85 | -3.91/0.114 | NA | -3.83 | NA | NA | NA |

D12S391 | 0.574 | 0.623 | 3.13 | -3.23/0.123 | NA | -5.47/0.139 | -4.99/0.227 | -3.78 | NA |

D13S317 | 0.822 | 0.85 | 4.08 | -5.73/0.201 | -4.02 | NA | NA | NA | NA |

D16S539 | 1.14 | 0.765 | 3.13 | -3.79/0.147 | -4.4 | -7.06/0.246 | NA | NA | NA |

D17S1301 | 0.543 | 0.989 | 15.6 | -4.25/0.2 | NA | -3.87 | NA | NA | NA |

D18S51 | 0.968 | 0.778 | 2.41 | -3.69/0.101 | -6.74/0.16 | -6.02/0.11 | NA | NA | NA |

D19S433 | 1.12 | 0.204 | 3.55 | -3.74/0.102 | NA | NA | NA | NA | NA |

D1S1656 | 0.305 | 0.867 | 4.26 | -4.3/0.17 | NA | NA | NA | NA | NA |

D20S482 | 2.44 | 0.867 | 3.12 | -2.13 | -3.97 | -4.87 | NA | NA | NA |

D21S11 | 0.533 | 0.746 | 3.87 | -3.54/0.0843 | -3.61 | NA | -3.36 | NA | NA |

D22S1045 | 1.44 | 0.948 | 6.04 | -3.6/0.118 | -5.57/0.146 | -6.2/0.136 | NA | NA | NA |

D2S1338 | 1.21 | 0.446 | 1.85 | -2.63/0.0505 | NA | -5.37/0.0982 | -6.92/0.427 | -4.38 | -4.37 |

D2S441 | 1.14 | 0.901 | 3.39 | -3.56 | -4.9 | NA | NA | NA | -4.31 |

D3S1358 | 2.08 | 0.286 | 2.43 | -4.15/0.127 | -6.85/0.177 | -7.05/0.178 | NA | -4.69 | -4.83 |

D4S2408 | 0.993 | 0.892 | 4.01 | -5.27/0.208 | -6.86/0.23 | NA | NA | NA | NA |

D5S818 | 0.29 | 0.968 | 12.4 | -2.85 | NA | NA | NA | NA | NA |

D6S1043 | 1.09 | 0.85 | 3.03 | -3.67/0.1 | -7.42/0.259 | -4.59 | NA | NA | NA |

D7S820 | 0.791 | 0.467 | 2.24 | -5.17/0.222 | -4.35 | NA | NA | NA | NA |

D8S1179 | 1.48 | 0.875 | 3.21 | -3.39/0.122 | -6.83/0.217 | -6.3/0.178 | NA | NA | NA |

D9S1122 | 1.55 | 0.812 | 3.46 | -4.77/0.207 | -4.19 | -7.53/0.231 | NA | NA | NA |

FGA | 1.05 | 0.655 | 3.56 | -3.03/0.0873 | -5.78/0.112 | -5.79/0.135 | NA | NA | NA |

PENTA E | 0.256 | 0.968 | 6.17 | -2.5 | NA | NA | NA | NA | NA |

TH01 | 2.77 | 0.728 | 2.97 | -4.19/0.193 | -4.81 | -5 | -5.25 | NA | NA |

TPOX | 0.894 | 0.938 | 3.81 | -5/0.167 | NA | NA | NA | NA | NA |

VWA | 0.382 | 0.958 | 3.09 | -3.56/0.12 | NA | NA | NA | NA | NA |

PENTA D | 0.181 | 0.431 | 2.83 | -5.7/0.189 | NA | NA | NA | NA | NA |

**A**is the estimated marker amplification efficiency,

**p**is geometrical model parameter for the number of noise alleles and

**lambda**is pareto model parameter for the noise size. The remaining columns indicate the stutter coefficients (intercept and slope) of the different stutter types separated by a backslash. Slope was only included if significant. “NA” means that the stutter type was not modelled.

### 3.2 LR for ${H}_{p}\phantom{\rule{1em}{0ex}}$true comparisons

### 3.3 ROC analysis and LR for ${H}_{d}\phantom{\rule{1em}{0ex}}$true comparisons

Thresh | Model | TPR | FPR | FN | FP |
---|---|---|---|---|---|

1 | EFM (T = 11) | 0.931 | 0.0238 | 11 | 10 |

1 | EFM (T = 20) | 0.969 | 0.0548 | 5 | 23 |

1 | EFM (T = 30) | 0.938 | 0.0905 | 10 | 38 |

1 | NB | 0.938 | 0.0857 | 10 | 36 |

1 | GA | 0.956 | 0.157 | 7 | 66 |

10 | EFM (T = 11) | 0.906 | 0 | 15 | 0 |

10 | EFM (T = 20) | 0.912 | 0 | 14 | 0 |

10 | EFM (T = 30) | 0.869 | 0 | 21 | 0 |

10 | NB | 0.906 | 0.00714 | 15 | 3 |

10 | GA | 0.944 | 0 | 9 | 0 |

### 3.4 Goodness of fit

### 3.5 Reducing the analytical threshold for EuroForMix

## 4. Discussion

### 4.1 Summary and overall findings

- Agudo M.M.
- Aanes H.
- Roseth A.
- Albert M.
- Gill P.
- Bleka Ø.

*Forensic Sci. Int. Genet.*2022; 102728https://doi.org/10.1016/j.fsigen.2022.102728

*calibrateModel*function in the software is used to perform all necessary calibration steps.

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

- Bleka Ø.
- Just R.
- Le J.
- Gill P.

*Forensic Sci. Int. Genet.*2020; 48102319https://doi.org/10.1016/j.fsigen.2020.102319

- Benschop C.C.G.
- et al.

*Forensic Sci. Int. Genet.*2021; 52https://doi.org/10.1016/j.fsigen.2021.102489

### 4.2 Model considerations

#### 4.2.1 Stutters

#### 4.2.2 Noise

- Vilsen S.B.
- Tvedebrink T.
- Mogensen H.S.
- Morling N.

*Forensic Sci. Int. Genet. Suppl. Ser.*2015; 5: e416-e417https://doi.org/10.1016/j.fsigss.2015.09.165

- Benschop C.C.G.
- et al.

*Forensic Sci. Int. Genet.*2021; 52https://doi.org/10.1016/j.fsigen.2021.102489

#### 4.2.3 Thresholds and data filtering

#### 4.2.4 Marker amplification efficiencies

#### 4.2.5 Degradation

#### 4.2.6 Data format

- Jønck C.G.
- Qian X.
- Simayijiang H.
- Børsting C.

*Forensic Sci. Int. Genet.*2020; 48102331https://doi.org/10.1016/j.fsigen.2020.102331

#### 4.2.7 Model differences

### 4.3 Current implementation and future work

- Benschop C.C.G.
- et al.

*Forensic Sci. Int. Genet.*2021; 52https://doi.org/10.1016/j.fsigen.2021.102489

## 5. Conclusion

## Conflict of interest

## Acknowledgements

## Appendix A

### Mathematical description of the distributions

**Gamma distribution:**

**Negative binomial distribution** (re-parameterized version):

### Deriving the relation between size parameter and the coefficient of variation

**Computational sidenote**: Very large values of $\eta $ can cause computational issues which is solvable by restricting it below a value ${\eta}_{0}$. It can be derived that the restriction $\eta <{\eta}_{0}$ is equivalent to $\mu >{\left({\omega}^{2}-\frac{1}{{\eta}_{0}}\right)}^{-1}$. Hence smaller values of $\omega $ would require a large value of $\mu $. We used a value of ${\eta}_{0}=1e4$ in the C++ function which avoids the maximum likelihood optimization to crash using parallelization with OpenMP.

## Appendix B. Supplementary material

Supplementary material

Supplementary material

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