Thanks to ...

Motivation

Standard approach to GRN inference

• perturb system (drug, ...)
• xi∈RD state of single condition i, averaged over all cells
• no or few dynamic information

Single-cell based GRN inference

• take snapshot of evolving system
• xi∈RD 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) Var(X|Universe)<Var(X|Universe∖Y)

Transfer entropy Schreiber, Phys. Rev. Lett. (2000) Entropy(X|Y)<Entropy(X)

▷  uncertainty in pred. ≃ 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) Var(X|Y)=0 if ′Y is coupled to X′

  ▷  functional coupling

  ▷  applicable to deterministic (chaotic) system

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

▷  predict Xg′ given Xg:

    ˆXtg′|Mg=∑x_t′g∈Ntg"selects t′ based on Mg"⏞w(x_tg,x_t′g)xt′g′

CCM for snapshot data

▷  predict Xg′ given Xg:

    ˆXtg′|Mg=∑x_t′g∈Ntg"selects t′ based on Mg"⏞w(x_tg,x_t′g)xt′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)
  N∑i=1(xig−fg(xi))2

What is already there?

• ODE based model Oconce, Haghverdi, Müller & Theis, Bioinformatics (2015)
   ▷  edges using GENIE3, then optimize for ug(ˆx)
  ddtˆxg=αug(ˆx)−λˆxg,  p(D|θ)∝∏Tt=1exp(−(xtg−ˆxg(t,θ))22σ2)
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!