6. Generation of a mouse ovary cell atlas with single-cell sequencing 97
6.4. Outlook
by adding the correct entry forGas5 to the annotation and then performing alignment, filtering and counting again.
6.4. Outlook
The next part of data gathering was planned for the summer months of 2019, overlapping with thesis writing and my thesis defense. However, once all data is available, it will be possible to integrate the different time points and study perinatal ovary development in single cell resolution.
MERLoT is a novel trajectory inference method that can reconstruct complex lineage trees by smoothing a robust initial tree and embedding into gene expression space. The embedding step is what sets MERLoT apart from most other methods (for example TSCAN [76] and Slingshot [161], which connect cell clusters and refine this rough tree), translating the lineage tree to gene expression space, facilitating downstream analysis, and improving the interpolated expression profiles of the lineage tree nodes. As the transcriptomics field continues advancing, bigger and more complex datasets are generated. MERLoT has already demonstrated, on real [45] and simulated [123] data, that it is well-equipped to deal with these challenges.
The last years have also produced significant advances in the field of visualization and dimensionality reduction. General purpose methods like PHATE [115] and UMAP [112] have been adapted while methods like URD [41] and PAGA [193] have been developed specifically for single cell transcriptomics visualization. While more conceptual representations like PAGA are useful, they might not be very well-suited to represent transitions, such as expression gradients within a cell type, in the same way that methods that visualize on the cell level can. MERLoT is not tied to a specific dimensionality reduction method, and can be applied on top of any coordinates, but it also offers conceptual visualizations for trees in higher dimensions.
Most dimensionality reduction algorithms and visualization techniques translate transcriptome sim-ilarity to distance on a reduced manifold. The quality of the mapping is directly related to how well these similarities are quantified. The usual choice, Euclidean distances on count or transformed data, suffer from the curse of dimensionality. A probabilistic distance kernel that uses an appropriate statis-tical model to represent the data would overcome this problem. We proposed a Gaussian mixture of negative binomial distributions to describe single cell count data from differentiation experiments. Us-ing this model to define a distance kernel is straightforward, but learnUs-ing the model hyperparameters is not (also see Risso et al. [136], who train a similar model). While negative binomial distributions have long been considered an appropriate model for count data [51, 72], they have not received as much attention in the computational side. The recent success of methods that use negative bino-mial distributions [130, 136, 38] and the proposal of a mechanistic transcription model [7] signal that maybe this trend is at an end, and soon we might see negative binomial kernels used in diffusion maps, UMAP, or t-SNE.
I am currently working on two approaches that don’t make explicit model assumptions. First, we
7. Conclusion and outlook 101 aim to decrease noise by weighted local averaging. Instead of only imputing missing values [97] or building a model and applying it to each cell [69], we transfer information from nearby cells, a much milder form of imputation. Since we don’t correct each gene separately we preserve the non-linear relationships between the genes. This approach is also more robust to the choice of distance measure;
if inappropriate cells are suggested as neighbours they will just receive very low contribution weights.
Second, we are interested in comparing our imputation procedure and others [38, 186, 69, 97, 179]
using an adaptation of cross validation for scRNA-seq. In particular, we want to use the robustness of trajectory inference as a measure of imputation quality. This would be a departure from established practice. To date, imputation quality is usually demonstrated by introducing zeros or noise in an expression matrix and trying to recover the original expression values [38] or comparing dimensionality reduction and gene expression plots before and after imputation.
A major development in the field of lineage reconstruction was the introduction of lineage tracing for developmental processes. Various techniques can be used to tag cells in developing embryos with random CRISPR-Cas9 scars starting at the zygote, so that entire cell populations can later be traced back to their ancestor cell via their scar barcodes [157, 83, 113]. Lineage tracing with CRISPR has not been widely adopted as of yet, but it is easy to imagine how much it could benefit trajectory inference. Achieving regular and predictable CRISPR scarring, e.g. once every cell division, would constitute an internal differentiation timer, providing a natural alternative to pseudotime.
Without such foolproof internal consistency markers for pseudotime or cell identity, the only way to validate method performance quantitatively is to use simulated data. In the early years of single-cell transcriptomics a lot of simulation methods were developed. They aimed to simulate different groups of cells with differential expression between them and account for other factors, such as dropout and batch effect [102, 101, 177]. Splatter [200] was a more sophisticated approach that included previous models, and could also simulate trajectories. However, the trajectory module of Splatter produced artifacts, as we discovered during the benchmarking of MERLoT [123]. PROSSTT addresses the need for simulated data of complex differentiation trajectories. Its flexible variance model covers a variety of different count distributions proposed for scRNA-seq data, and its model of gene expression change over pseudotime creates rich correlations between the different genes, mimicking the relationships between genes in real data. We think PROSSTT fills an important niche, and can be a valuable tool for the development of trajectory inference tools.
The increase in scRNA-seq throughput has brought the once lofty goal of dissecting gene regulatory networks within reach [55] aided considerably by the advances in single cell epigenetics, and methods to infer networks from single-cell data are being developed [114, 21]. Method development would be greatly facilitated by the existence of data with known regulatory relationships, real or simulated, and while it is possible to simulate simpler regulatory networks and the corresponding thermodynamics [142], this approach doesn’t scale beyond a small set of predetermined topologies. Integrating gene regulation to PROSSTT could be an interesting expansion.
Single cell transcriptomics has gone from an experimental breakthrough to widely applied standard technique in the span of ten years. This was accompanied by corresponding advances in computational analysis. Yet, despite the progress, there is still room for improvement: challenges like noise modelling, batch effect removal, and trajectory inference, particularly on large datasets, remain, while new ones, like incorporation of spatial information, and integration of data from multiple sources and data of
Appendices
the current standard in the field right now. Suspiciously round numbers are not exact but correspond to the authors’ descriptions of the datasets.
Year Month #Cells Citation
2009 5 1 Tanget al. [163]
2010 5 34 Tanget al. [164]
2011 5 85 Islamet al. [71]
2011 6 24 Tanget al. [165]
2012 8 96 Hashimshonyet al. [64]
2012 8 12 Ramsk¨old et al. [132]
2012 10 6 Brouiletteet al.[19]
2013 4 77 Sasagawaet al. [143]
2013 4 10 Magn´usd´ottiret al. [105]
2013 6 18 Shaleket al. [150]
2013 7 124 Yanet al. [198]
2013 11 100 Brenneckeet al. [17]
2013 11 68 Picelliet al. [126]
2013 12 18 Grindberget al. [50]
2013 12 15 Marinovet al.[108]
2014 1 100 Wuet al.[195]
2014 2 192 Islamet al. [72]
2014 2 4,000 Jaitinet al. [73]
2014 5 400 Treutleinet al. [174]
2014 6 1,800 Shaleket al. [151]
2014 6 768 Patelet al. [124]
2014 12 415 Kumaret al. [92]
2015 2 15,000 Fanet al.[39]
2015 4 470 Darmaniset al. [29]
2015 4 3,000 Zeiselet al. [202]
2015 5 44,808 Macoskoet al. [104]
A. Representative single-cell sequencing methods in the last ten years II 2015 5 9,000 Kleinet al. [89]
2015 5 400 Boseet al. [15]
2015 10 869 Kolodziejczyket al. [90]
2015 11 9,000 Paulet al. [125]
2016 3 1,682 Habibet al. [57]
2016 4 4,600 Tiroshet al. [171]
2016 6 5,000 Marqueset al. [109]
2016 6 3,000 Lakeet al. [95]
2016 7 1,800 Tasicet al. [167]
2016 9 2,200 Yuan and Sims [199]
2016 9 1,422 Joostet al. [81]
2016 10 3,000 Vickovicet al. [183]
2016 10 6,100 La Mannoet al. [93]
2016 10 12,200 Muraroet al. [116]
2016 12 200,000 Dixitet al.[33]
2016 12 50,000 Jaitinet al. [74]
2016 12 86,000 Adamsonet al. [2]
2017 1 250,000 Zhenget al.[205]
2017 1 5,905 Datlingeret al. [31]
2017 1 1,369 Alleset al. [5]
2017 2 4,638 Gierahnet al. [46]
2017 2 42,000 Caoet al. [24]
2017 2 1,500 Halpernet al. [61]
2017 3 1,486 Guillaumet-Adkinset al.[54]
2017 3 14,391 Venteicheret al.[182]
2017 3 14,000 Chenet al. [25]
2017 4 2,400 Villaniet al. [184]
2017 6 2,167 Liet al. [96]
2017 7 2,616 Stoeckiuset al. [160]
2017 8 29,543 Habibet al. [58]
2017 8 20,424 Adamet al. [1]
2017 10 3,589 Darmaniset al. [30]
2017 10 20,679 Wuet al.[197]
2017 10 7,975 Karaiskoset al. [86]
2017 11 18,000 Poranet al.[129]
2017 11 53,193 Haberet al.[56]
2017 12 11,759 Schelkeret al.[145]
2017 12 18,000 Huet al. [68]
2017 12 14,104 Kanget al. [84]
2018 1 5,454 Hochgerneret al. [67]
2018 1 10,519 Skellyet al. [156]
2018 3 3,655 Shnayderet al. [154]
2018 4 156,049 Rosenberget al. [138]
2018 4 2458 Filbinet al. [43]
2018 4 25,000 van der Wijstet al. [178]
2018 4 13,000 Pandeyet al. [120]
2018 4 2,880 Hermanet al. [66]
2018 4 10,000 Alemanyet al. [4]
2018 5 11,888 Seb´e-Pedr´oset al. [148]
2018 5 15,539 Roddaet al. [137]
2018 5 25,790 Nguyenet al. [118]
2018 5 57,979 Parket al. [122]
2018 5 6,862 Kimet al. [87]
2018 5 21,612 Plasset al. [128]
2018 6 90,000 Wagneret al.[185]
2018 6 38,731 Farrellet al.[42]
2018 6 4,000 Fanet al.[40]
2018 6 136,966 Briggset al. [18]
2018 7 3,142 Wanget al. [190]
2018 7 10,000 Sasagawaet al. [144]
2018 8 34,188 Andoret al. [9]
2018 8 509,876 Zeiselet al. [203]
2018 9 1,500 Karaayvazet al. [85]
2018 9 35,000 Greenet al. [49]
2018 12 1,976 Collinet al. [28]
2018 12 76,070 Reyfmanet al. [135]
2019 1 76,000 Hammondet al. [62]
2019 1 1,464 Jiet al. [75]
2019 1 3,461 Jord˜aoet al. [82]
2019 1 36,931 Bragaet al.[16]
2019 2 1,106 Tiklov´a et al. [170]
2019 2 10,000 Ryuet al. [141]
2019 2 315,000 Schiebingeret al. [146]
2019 3 38,410 van Galenet al. [180]
2019 3 3,152 Francesconiet al. [44]
2019 4 1,088 Wenet al. [192]
2019 4 2,607 Sharmaet al.[152]
A. Representative single-cell sequencing methods in the last ten years IV 2019 4 17,374 Tikhonovaet al. [169]
2019 5 12,000 Shulseet al. [155]
since the derivatives of the logs of all probabilities to the right ofp(ρ) are constant per definition, we only need the following derivatives:
p(s|σs) → sn, σs
B. Partial derivatives of posterior probability for noise model VI
g=1k=1
∂
∂βlogp(µ|α, β) =KGα β −
XG g=1
XK k=1
µkg (B.10)
∂
∂µkglogp(µ|α, β) = ∂
∂µkg XG g=1
XK k=1
(αlogβ+ (α−1) logµkg−µkgβ−log Γ(α))
= ∂
∂µkg(αlogβ+ (α−1) logµkg−µkgβ−log Γ(α))
= (α−1)
µkg −β (B.11)
∂
∂µkg logp(µ|α, β) = (α−1)
µkg −β (B.12)
B. Partial derivatives of posterior probability for noise model VIII
∂bg 2 b0bg a0bg
B. Partial derivatives of posterior probability for noise model X
=
Again, it is not necessary to calculate everything since not all variables appear. The partial derivatives that are needed are the ones with respect toρk, cn, µkg, ag, bg.
In the calculations the following notation is used to improve readability and make the calculations easier:
kg−mkg, pkg being the probability of success and rkg the predefined number of failures, both parametrized by the mean and variance of the distribution.
B. Partial derivatives of posterior probability for noise model XII
= contain the varianceσkg2 , since only those contain cn.
log NegBin(xng|rng, pkg) = log before going on with the calculations we need to calculate the derivatives of 1−pkg and rkg, since we will need them a lot. In this case we can ignore mkg as it doesn’t containcn.
B. Partial derivatives of posterior probability for noise model XIV
and now continuing from eq. B.37:
+ log(1−pkg)
∂cnrkg+rkg
(1−pkg)∂cn(1−pkg) +xng
pkg∂cnpkg
= −ψ0(xng+rkg)rkg pkg
1
cn +ψ0(rkg)rkg pkg
1
cn+ log(1−pkg)rkg pkg
1 cn rkg 1
(1−pkg)
pkg−1
cn +xng 1 pkg
1−pkg cn
= 1
cn rkg
pkg(ψ0(rkg)−ψ0(xng+rkg) + log(1−pkg))
−rkg+xng
1−pkg pkg
(B.41) The same procedure can be followed when deriving bysn, as it only occurs inside the product.
The general form of the derivative is the same (equations B.36 and B.37). We recalculate ∂r and ∂(1−pkg), this time following the terms that contain mkg.
B. Partial derivatives of posterior probability for noise model XVI
∂rkg
∂sn = ∂
∂sn
m2kg σ2kg−mkg
= m2kg0(σkg2 −mkg)−(σkg2 −mkg)0m2kg (σkg2 −mkg)2
= 2snµ2kg(σkg2 −mkg)−s2nµ2kg(cnags2nµ2kg+cnbgsnµkg−snµkg)0 (σ2kg−mkg)2
= 2snµ2kg(σkg2 −mkg)−s2nµ2kg(2cnagsnµ2kg+cnbgµkg−µkg) (σ2kg−mkg)2
= 2snµ2kg(σkg2 −mkg)−snµ2kg(cnags2nµ2kg+cnbgsnµkg−snµkg+cnags2nµ2kg) (σkg2 −mkg)2
= 2snµ2kg(σkg2 −mkg)−snµ2kg(σkg2 −mkg+cnags2nµ2kg) (σ2kg−mkg)2
= snµ2kg(σ2kg−mkg−cnags2nµ2kg) (σkg2 −mkg)2
= snµ2kg(cnags2nµ2kg+cnbgsnµkg−snµkg−cnags2nµ2kg) (σ2kg−mkg)2
= snµ2kg(cnbgsnµkg−snµkg) (σkg2 −mkg)2
= s2nµ3kg(cnbg−1) (σkg2 −mkg)2
= s2nµ2kg σkg2 −mkg
µkg(cnbg−1) σkg2 −mkg
= rkgµkg(cnbg−1) σkg2 −mkg
(B.42)
= µkgσ2kg−snµkg(2cnagsnµ2kg+cnbgµkg) For the derivatives with respect toag,bg andµkg the overall structure is the same. Here we get to keep the sum over all cells (n) but since the derivatives for g0 6=g are 0 we do not need the product rule:
B. Partial derivatives of posterior probability for noise model XVIII For the rest of the calculations we proceed in a similar matter; we need the derivatives of 1−pkg, rkg. In the case of ag (and laterbg) we are only interested in terms containing σkg2 .
= −mkg
The partial derivative with respect to bg is virtually identical, with the exception of using µkg instead ofµ2kg.
B. Partial derivatives of posterior probability for noise model XX other than that we proceed in the same fashion, and calculate the helper derivatives:
= 2µkgs2n(σ2kg−mkg)−(2s2nµkgcnag+bgsncn−sn)s2nµ2kg (σkg2 −mkg)2
= 2µkgs2n(σ2kg−mkg)−(2s2nµ2kgcnag+snµkgbgcn−snµkg)s2nµkg (σkg2 −mkg)2
= 2µkgs2n(σ2kg−mkg)−((s2nµ2kgcnag+snµkgbgcn)−snµkg+s2nµ2kgcnag)s2nµkg (σkg2 −mkg)2
= 2µkgs2n(σ2kg−mkg)−(σkg2 −mkg+s2nµ2kgcnag)s2nµkg (σkg2 −mkg)2
= µkgs2n(2(σkg2 −mkg)−(σ2kg−mkg)−s2nµ2kgcnag) (σ2kg−mkg)2
= µkgs2n(σ2kg−mkg−s2nµ2kgcnag) (σ2kg−mkg)2
= µkgs2n(s2nµ2kgcnag+µkgcnsnbg−snµkg−s2nµ2kgcnag) (σkg2 −mkg)2
= µ2kgs3n(cnbg−1) (σ2kg−mkg)2
= m2kg
(σkg2 −mkg)sn (cnbg−1) (σ2kg−mkg)
= rkgsn (cnbg−1)
(σkg2 −mkg) (B.56)
B. Partial derivatives of posterior probability for noise model XXII
• The derivative ofxeig, since it comes up in all formulas:
∂exig
∂wj = xjg
P
jwj − P
jwjxjg (P
jwj)2
= xjg−xeig P
jwj
∂exig
∂wj = xjg−exig P
jwj (C.1)
• The second helper derivative isσe2ig.
∂σe2ig
∂wj = (xig−exig)2−σe2ig P
jwj (C.2)
• The derivative of the bias:
∂bias
∂wj = 1 2P
jwj X
g
−2(xig−xeig)(xjg−xeig) e
σig2 −(xig−exig)2(xjg−exig)2 e
σ4ig + (xig −exig)2 e σig2
(C.3)
• and the derivative of the variance term:
∂eσig2
∂wj = Gwj (P
jwj)2 −G(P
jwj2) (P
jwj)3 (C.4)
35(1), 47–54.
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