Inference of gene regulation using pseudotemporal ordering of single cell snapshots

Thanks to ...

Motivation

Standard approach to GRN inference

• perturb system (drug, ...)
• $\boldsymbol{x}_i \in \mathbb{R}^D$ state of single condition $i$, averaged over all cells
• no or few dynamic information

Single-cell based GRN inference

• take snapshot of evolving system
• $\boldsymbol{x}_i \in \mathbb{R}^D$ state of single cell $i$
• dense but hidden dyn. information

• How can we reveal and exploit this?

Introduction: Correlation

• Correlation does not imply causation.
• Absence of correlation does not imply absence of causation.

example from Haghverdi, Buettner & Theis, Bioinformatics (2015)
data from M. Strasser

Introduction: Is $Y$ causal for $X$?

• Granger causality Granger, Econometrica (1969) $$ \text{Var}(X|\text{Universe}) < \text{Var}(X|\text{Universe}\backslash Y) $$

Transfer entropy Schreiber, Phys. Rev. Lett. (2000) $$ \text{Entropy}(X|Y) < \text{Entropy}(X)$$

▷  uncertainty in pred. $\simeq$ information flow Barnett et al., Phys. Rev. Lett. (2009)

▷  applicable to stochastic system

• Convergent cross mapping (CCM) Sugihara et al., Science (2012), Takens' Theorem (1980) $$ \text{Var}(X|Y) = 0 \text{ if } 'Y \text{ is coupled to } X' $$

  ▷  functional coupling

  ▷  applicable to deterministic (chaotic) system

CCM for temporal data Sugihara et al., Science (2012)

▷  predict $X_{g'}$ given $X_{g}$:

$$~~~~\hat X_{t g'} | M_{g} = {\textstyle\sum_{\scriptscriptstyle \underline{x}_{t'g}\in\, \mathcal{N}_{t g}}} \!\!\!\!\!\!\!\overbrace{w(\underline{x}_{tg},\underline{x}_{t'g})}^{\text{"selects } t' \text{ based on } M_g \text{"}} \!\!\!\!\!x_{t'g'}$$

CCM for snapshot data

▷  predict $X_{g'}$ given $X_{g}$:

$$~~~~\hat X_{t g'} | M_{g} = {\textstyle\sum_{\scriptscriptstyle \underline{x}_{t'g}\in\, \mathcal{N}_{t g}}} \!\!\!\!\!\!\!\overbrace{w(\underline{x}_{tg},\underline{x}_{t'g})}^{\text{"selects } t' \text{ based on } M_g \text{"}} \!\!\!\!\!x_{t'g'}$$

Bendall, Davis, ... Peer & Nolan, Cell (2014)
Trapnell, Cacchiarelli, ... Mikkelsen & Rinn, Nat. Biotechn. (2014)
Haghverdi, Büttner, ... Buettner & Theis, submitted (2016)

CCM for snapshot data

Straight-forward improvements

• use monotonicity / variance
• use relative predictivity and time lag  ▷  transitivity

Conceptual comparisons

• Transfer Entropy
• GENIE3: Random Forests
  
  Huynh-Thu, ..., Geurts, PLOS One (2010)
  $$ \sum_{i=1}^N (x_{ig} - f_g(\boldsymbol{x}_i))^2 $$

What is already there?

• ODE based model Oconce, Haghverdi, Müller & Theis, Bioinformatics (2015)
   ▷  edges using GENIE3, then optimize for $ u_g(\hat{\boldsymbol{x}})$
  $$\frac{d}{dt} \hat x_g = \alpha u_g(\hat{\boldsymbol{x}}) - \lambda \hat x_g,~~ \textstyle p(\mathcal{D}|\theta) \propto \prod_{t=1}^T \exp\big(-\frac{(x_{tg} - \hat x_g(t,\theta))^2}{2 \sigma^2}\big) $$
Discrete state space model Moignard, Woodhouse ... Goettgens, Nat. Biotech. (2015)
• generate discrete state graph of 1-gene transitions
• check consistency of trial Boolean networks

Summary

Convergent Cross mapping (CCM) Sugihara et al., Science (2012)

• detects "causality"=couplings in noisy deterministic systems
  (even without noise, such systems might seem stochastic when they are "merely" chaotic)
• is alternative to Transfer Entropy / Granger Causality
  (for "purely" stochastic systems)

Pseudotime + CCM for GRN inference

• pseudotime: reveals structure of manifold associated with "deterministic part" of a GRN (i.e. a dynamic system)
• peudotime+CCM: qualitatively better than naive approaches
• many questions still open ... ☺

Thank you!