1. Computational and Systems Biology
  2. Chromosomes and Gene Expression
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Stochastic modelling, Bayesian inference, and new in vivo measurements elucidate the debated mtDNA bottleneck mechanism

  1. Iain G Johnston
  2. Joerg P Burgstaller
  3. Vitezslav Havlicek
  4. Thomas Kolbe
  5. Thomas Rülicke
  6. Gottfried Brem
  7. Jo Poulton
  8. Nick S Jones  Is a corresponding author
  1. Imperial College London, United Kingdom
  2. IFA Tulln, Austria
  3. University of Veterinary Medicine Vienna, Austria
  4. University of Veterinary Medicine, Austria
  5. University of Natural Resources and Life Sciences, Austria
  6. University of Oxford, United Kingdom
Research Article
Cite this article as: eLife 2015;4:e07464 doi: 10.7554/eLife.07464
10 figures and 3 tables

Figures

The mitochondrial bottleneck, and elements of a general model for bottlenecking mechanisms.

(A) The mitochondrial DNA (mtDNA) bottleneck acts to produce a population of oocytes with varying heteroplasmies from a single initial oocyte with a specific heteroplasmy value. During development, mtDNA copy number per cell decreases (by a debated amount, which we address; see Main text) then recovers, suggesting a ‘bottleneck’ of cellular mtDNA populations. (B) Cellular mtDNA populations during the bottleneck are modelled as containing wildtype and mutant mtDNAs. MtDNAs can replicate and degrade within a cell cycle, with rates λ and ν respectively. (C) At cell divisions, the mtDNA population is partitioned between two daughter cells either deterministically, binomially, or through the binomial partitioning of mtDNA clusters. (D) Symbols used to represent quantities and model parameters used in the Main text, and their biological interpretations.

https://doi.org/10.7554/eLife.07464.003
Different mechanisms for the mtDNA bottleneck.

(A) Trajectories of mean copy number E(m) and normalised heteroplasmy variance V(h) arising from the models described in the text, optimised with respect to data from experimental studies. Birth-death-partition (BDP) denotes the BDP model, encompassing Cree and Wai (A) mechanisms. Left plots show trajectories during development; right plots show behaviour in mature oocytes in the next generation. * denotes measurements in mature oocytes, modelled as 100 dpc (see ‘Materials and methods’). (B) Statistical support for different mechanisms from approximate Bayesian computation (ABC) model selection with thresholds ϵ1,2,3,4 = 75, 60, 50, 45. As the threshold decreases, forcing a stricter agreement with experiment (thinner, darker columns), support converges on the BDP model.

https://doi.org/10.7554/eLife.07464.004
Parameterisation of the BDP model and inferred details of bottleneck mechanism.

Trajectories of (A) mean copy number E(m) and (B) normalised heteroplasmy variance V(h) resulting from BDP model parameterisations sampled using ABC with a threshold ϵ = 40. * denotes measurements in mature oocytes, modelled as 100 dpc (see ‘Materials and methods’). Note: the range in (B) does not correspond to a credibility interval on individual measurements, but rather on an expected underlying (population) variance, from which individual variance measurements are sampled. We thus expect to see, for example, several measurements lower than this range due to sampling limitations (see text). (C) Posterior distributions on mtDNA turnover ν with time. (D) Posterior distribution on min E(m), the minimum mtDNA copy number reached during development. (E) Posterior distribution on σ=i=36τiνi, a measure of the total amount of mtDNA turnover.

https://doi.org/10.7554/eLife.07464.005
Predictions and experimental verification of the BDP model.

(A) New V(h) measurements from the HB mouse system, with optimised fits for the BDP, Wai (b) and Cao models. (B) Posterior probabilities of each model given this data under decreasing ABC threshold: ϵ = {50, 40, 30, 25}. (C) All V(h) measurements from the HB model (points) with inferred V(h) behaviour from ABC applied to the BDP model (red curves). As in Figure 3, this range does not correspond to a credibility interval on individual measurements, but rather on an expected underlying (population) variance, from which individual variance measurements are sampled. The inferred behaviour strongly overlaps with the inferred behaviour for the BALB/c system (blue curves), suggesting that the BDP model applies to a genetically diverse range of systems. (D) Heteroplasmy distributions. The transformation h=ln|(h11)E(h)/(1E(h))| (Burgstaller et al., 2014) is used to compare distributions with different mean heteroplasmy. Red jitter points are samples from sets used to parameterise the BDP model; red curves show the 95% range on transformed heteroplasmy with time inferred from these samples. Blue jitter points are samples withheld independent from this parameterisation; their distributuions fall within the independently inferred range. Insets show, in untransformed space, distributions of the withheld heteroplasmy measurements (blue) compared to parameterised predictions (red); no withheld datasets show significant support against the predicted distribution (Anderson-Darling test, p < 0.05).

https://doi.org/10.7554/eLife.07464.006
Figure 4—source data 1

Individual heteroplasmy measurements in the HB mouse model contributing to the new heteroplasmy variance data used to test our theory.

https://doi.org/10.7554/eLife.07464.007
Quantitative influences and clinical results from our bottlenecking model.

(AC) Trajectories of copy number E(m) and normalised heteroplasmy variance V(h) resulting from perturbing different physical parameters. Trajectory C labels the ‘control’ trajectory resulting from a fixed parameterisation; black dots show experimental data; * denotes measurements from primary oocytes, modelled at 100 dpc. (A) Increasing (T+) and decreasing (T) mtDNA turnover (both mtDNA replication and degradation) by 20%. (B) Increasing (M+) and decreasing (M) mtDNA degradation throughout development by a constant value (2 × 10−4, in units of day−1), while keeping replication constant. (C) Applying a positive (S+) and negative (S) selective pressure to mutant mtDNA by 5 × 10−6 day−1. (D) Probability of crossing different heteroplasmy thresholds h* with time, starting with initial heteroplasmy h0 = 0.3. (E) Probability distributions over embryonic heteroplasmy h given a measurement hm from preimplantation sampling (** hm = 0.1; *** hm = 0.4) at different times.

https://doi.org/10.7554/eLife.07464.008
Model for the mtDNA bottleneck.

A summary of our findings. (A) There is most statistical support for a bottlenecking mechanism whereby mtDNA dynamics is stochastic within a cell cycle, involving random replication and degradation of mtDNA, and mtDNAs are binomially partitioned at cell divisions. (B) This mechanism results in heteroplasmy variance increasing both due to stochastic partitioning at divisions and due to random turnover. The absolute magnitude of the copy number bottleneck is not critical: a range of bottleneck sizes can give rise to observed dynamics. Random turnover of mtDNA increases heteroplasmy variance through folliculogenesis and germline development.

https://doi.org/10.7554/eLife.07464.009
Appendix figure 1
Heteroplasmy variance in a model system under several different group-inheritance regimes.

V(h) over many cell divisions when the elements of inheritance are heteroplasmic or homoplasmic groups of different size. Groups may be quenched (Q; constituents remain the same across cell divisions) or unquenched (UQ; constituents are randomly resampled from the cellular population each cell cycle); for homoplasmic clusters, an unquenched protocol yields identical results to the quenched protocol. V(h) behaviour differing from binomial partitioning (c = 1) is only observed for homoplasmic groups with c ≥ 2. Points for heteroplasmic groups are slightly offset in the x-direction for clarity.

https://doi.org/10.7554/eLife.07464.012
Appendix figure 2
Residual distributions at different ABC thresholds ϵ.

The distribution of squared residuals corresponding to individual experimental datapoints compared to an ensemble of simulated trajectories for (top) logE(m) (bottom) V(h). The V(h) residuals are scaled by A1 = 103 to ensure that the two sets of measurements are compared on a quantitatively equal footing. As ϵ is decreased (ϵ1,2,3,4 = 40, 50, 75, 100), distributions of residuals from accepted trajectories tighten around zero.

https://doi.org/10.7554/eLife.07464.013
Appendix figure 3
Posterior distributions on model parameters.

The posterior distributions on individual model parameters, assuming the inferred BDP bottlenecking mechanism. Replication rates are presented as κ = λν, thus representing overall proliferation rates of mtDNA. Units are omitted for clarity. Pale, single-values distributions correspond to parameter values fixed within the model (κ6 = 0 to prevent mtDNA proliferation after development; τ1 = 7 hr, τ2 = 16 hr fixed by data on cell doubling times; h0 = 0.2 fixed for simplicity as heteroplasmy variances are normalised; δλ = 0 fixed to avoid varying selective pressure). The ‘turnover’ parameter, described in the text, is i=36τiνi, a measure of the total random turnover in the mtDNA population.

https://doi.org/10.7554/eLife.07464.014
Appendix figure 4
Comparison of truncated Normal approximation with exact heteroplasmy distribution.

Representations of heteroplasmy distributions at a time t = 21 dpc, with various starting heteroplasmies, using (as an example) the maximum likelihood parameterisation emerging from the inference procedure in the Main text. Dark lines and bars show exact distributions from Equation 72; pale lines and bars show distributions arising from the truncated Normal distribution described in the text.

https://doi.org/10.7554/eLife.07464.016

Tables

Table 1

Source data used in this study

https://doi.org/10.7554/eLife.07464.010
Time/dpcE(m)NSource study
02.5e522Cree
0.292.5e518Cree
0.585.8e49Cree
0.736.3e419Cree
0.884.8e433Cree
1.311.8e411Cree
7.54.5e2596Cree
8.51.1e3165Cree
10.51.6e396Cree
14.51.8e32615Cree, §
01.7e542Cao
0.297.1e432Cao
0.584.8e432Cao
0.882.1e432Cao
5.51.5e385Cao, §
6.51.6e343Cao, §
7.52.0e353Cao, §
7.752.0e342Cao, §
8.52.1e382Cao, §
9.52.2e393Cao, §
10.52.1e374Cao, §
11.52.0e367Cao, §
12.51.9e3124Cao, §
13.52.1e371Cao, §
8.52.8e220Wai#
9.52.5e320Wai#
10.52.8e320Wai#
12.54.0e320Wai#
14.55.8e320Wai#
16.52.4e320Wai#
255.0e320Wai#
291.0e420Wai#
323.0e420Wai#
466.5e420Wai#
Time/dpcV(h)NSource study
7.55.2e-612Jenuth
7.50.0084Jenuth
7.50.0043Jenuth
7.50.0085Jenuth
230.03940Jenuth
230.03237Jenuth
230.04035Jenuth
230.03835Jenuth
230.03734Jenuth
230.02148Jenuth
230.02645Jenuth
*0.14026Jenuth, **
*0.01724Jenuth, **
*0.14431Jenuth, **
*0.02149Jenuth, **
*0.03331Jenuth, **
8.50.01620Wai††, ‡‡
8.50.01820Wai††, ‡‡
9.50.01820Wai††, ‡‡
10.50.01720Wai††, ‡‡
10.50.02520Wai††, ‡‡
10.50.00820Wai††, ‡‡
12.50.01720Wai††, ‡‡
13.50.01420Wai††, ‡‡
13.50.01520Wai††, ‡‡
13.50.02020Wai††, ‡‡
13.50.02120Wai††, ‡‡
14.50.00420Wai††, ‡‡
16.50.01520Wai††, ‡‡
25.00.00220Wai††, ‡‡
25.00.00220Wai††, ‡‡
29.00.01320Wai††, ‡‡
29.00.02720Wai††, ‡‡
32.00.01620Wai††, ‡‡
32.00.02320Wai††, ‡‡
35.00.02620Wai††, ‡‡
35.00.02920Wai††, ‡‡
50.00.03920Wai††, ‡‡
51.10.04320Wai††, ‡‡
65.00.02620Wai††, ‡‡
  1. Data referenced by number of cells post–conception is assigned a time measurement assuming the 7 hr → 16 hr doubling times from Lawson and Hage (1994).

  2. Mean copy number taken directly from tabulated data.

  3. §

    (Weighted) average over germline cell classes presented at this time point.

  4. #

    Extracted from data in figures; n not explicitly available so estimated as n = 20 from accompanying histograms and discussion.

  5. Manually normalised from given data.

  6. **

    Data from mature oocytes in next generation: time in dpc not available.

  7. ††

    Extracted from data in figure in correspondence following study.

  8. ‡‡

    n not explicitly available so estimated as n = 20 from accompanying histograms and discussion in original paper.

Table 2

New heteroplasmy measurements from the HB model system

https://doi.org/10.7554/eLife.07464.011
Age3344448899924373740
n253021131311303420173625242020
E(h)0.5010.4190.1830.3370.3820.3540.3010.5590.1930.2450.0490.4570.5660.2760.238
V(h)0.002560.003590.008240.013080.009130.004610.003500.006310.004080.003010.000970.006250.010000.009130.00662
V(h)0.01020.01470.05510.05850.03870.02020.01670.02560.02620.01630.02100.02520.04070.04570.0364
h × 10040.826.57.416.726.925.118.238.78.313.31.732.538.012.88.9
43.430.67.623.128.825.722.041.79.316.91.732.639.213.812.1
44.131.87.924.029.429.123.743.910.820.11.837.145.915.913.9
44.233.29.624.429.830.823.946.413.520.21.937.150.916.615.7
46.636.49.726.830.236.524.647.513.520.22.038.051.020.318.8
46.737.710.327.036.736.725.947.915.821.32.839.551.720.619.1
46.937.912.428.637.138.226.049.516.323.72.840.051.721.819.5
47.438.714.339.840.238.326.251.416.424.92.941.151.823.620.8
48.039.214.840.440.540.426.351.518.425.22.943.052.023.822.5
48.439.516.041.943.643.626.952.818.726.23.043.754.126.222.7
48.539.717.042.945.844.826.952.820.727.03.043.754.228.323.2
48.741.417.150.146.727.852.920.727.33.044.254.429.623.4
49.342.118.652.860.528.953.221.327.63.044.554.932.627.8
50.342.619.729.153.221.827.83.346.355.333.128.9
50.542.720.829.753.523.828.23.346.655.733.129.4
50.642.725.729.953.924.230.03.547.057.234.530.1
50.842.925.730.154.226.536.73.549.157.538.430.9
51.243.826.430.755.226.53.749.859.940.534.8
53.744.528.331.356.026.83.850.161.142.235.2
54.544.735.631.756.232.13.851.864.843.939.4
55.044.839.332.657.14.053.069.9
56.245.933.057.34.054.374.1
56.647.033.459.84.955.776.1
59.047.633.660.05.555.776.5
62.048.734.760.15.866.2
48.734.961.35.9
48.835.361.36.0
48.935.862.16.1
49.141.265.66.7
49.748.567.17.9
68.38.2
69.28.3
69.48.5
70.18.8
11.6
16.6
  1. Heteroplasmy measurements and statistics from the HB model system. Ages are given in days after birth.

Table 3

Experiments for further elucidation of the mtDNA bottleneck

https://doi.org/10.7554/eLife.07464.015
MeasurementPurpose
MtDNA copy number before and after cell divisions and/or variance of copy number between daughter cellsTo elucidate mechanism of mtDNA partitioning and whether this partitioning is deterministic or stochastic
Copy number trajectories with different mtDNA heteroplasmiesTo assess the modulation of copy number dynamics by mtDNA heteroplasmy via retrograde signalling
Measurement of mean heteroplasmy through development, with a variety of mtDNA type pairingsTo assess and quantify to what extent selection modulates mtDNA dynamics during germline development
Copy number measurements after upregulation of mitophagyTo assess the presence and strength of compensatory mechanisms that may act to preserve mtDNA copy number—and hence whether upregulating mitophagy will act to increase mtDNA turnover or simply lower copy number
Heteroplasmy variance after upregulation of mitophagyTo assess the efficacy of mitophagy for increasing the power of the bottleneck
Heteroplasmy distribution in cells after the bottleneck from sampled/known initial heteroplasmyTo confirm predictions for threshold crossing and statistics between generations
BrU incorporation in oocytes between 30 and 40 dpcTo confirm the random turnover mechanism: we expect a large proportion of BrU incorporation subset of mtDNAs to be observed in this time period (see section ‘Observation of a subset of replicating genomes’)
Mitochondrial ultrastructure and mtDNA localisation during developmentTo assess and characterise any potential modulation of the size of units of mitochondrial inheritance by mitochondrial dynamics through development, in particular, investigating whether there is time-varying modulation of cluster size at points of division

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