Computing correspondences in order to study spatial and temporal patterns of gene expression

1. Computing correspondences in order to study spatial and temporal patterns of gene expression Charless Fowlkes UC Berkeley, Computer Science

2. Why do we need correspondence? • How can we identify corresponding nuclei in two different embryos? • What does it mean for a correspondence to be correct?

3. Why do we need correspondence?

4. Why do we need correspondence? Correspondence allows us compare the expression of a given gene across different embryos of the same stage.

5. Correspondence allows us build composite maps which show expression for multiple genes. Important for high thruput, N images rather than N2 Why do we need correspondence?

6. Why do we need correspondence? Correspondence allows us to compare gene expression patterns between embryos at different time points

7. How can we find corresponding nuclei?

8. How do we find corresponding nuclei? • Traditional approach: nuclei at the same percent egg length correspond. • Coarse Alignment: • align the principal axis of each embryo • scale to the same egg length • select the appropriate d/v orientation Nuclei with the same a/p and d/v coordinates correspond

9. FTZ average after coarse alignment

10. Detailed correspondence based on morphology alone appears quite difficult

11. Solution: • Use a reference gene expression pattern to identify corresponding keypoints (nuclei on the edge of an eve or ftz stripe) • Extend this correspondence to the non-expressing cells using a smooth coordinate transformation Underlying assumption is that nuclei are identified by expression.

12. Cij = difference in local expression pattern for points i and j Dij = distance between points i and j minimize : Σij (Cij + λDij) • Xij subject to : ΣiXij = 1 Σj Xij = 1 λ sets the relative importance of distance versus expression similarity match Correspondence as optimization Xij = 1 if point i is matched to point j 0 otherwise i j

13. Problem: correspondence may not be smooth • Find correspondence by optimizing Xij • Smoothly warp source embryo to bring into alignment with corresponding points • Repeat… Solution: iteratively correspond and warp

16. FTZ average after coarse alignment FTZ average after detailed correspondence

17. Building a composite expression map X1 X2 Push expression levels forward thru correspondence function X

18. How about correspondence between different time points?

19. Idea: use the average shape combined with average nuclear density to estimate motions

20. Generating synthetic time-series • Place 6000 nuclei in 3D such that they have the average shape and density pattern of early stage 5 blastoderm • Find a motions for all synthetic nuclei which yield a new set of 3D locations that match the average shape and density of late stage 5 blastoderm Regularization: seek the smallest, smoothest motion which fits the densities Optimizaiton: conjugate gradient

22. Predicted nuclear movements

23. What does it mean for a correspondence to be correct?

24. FTZ average after coarse alignment FTZ average after detailed matching More flexible warping results in sharper reference pattern but poorer prediction about location of other genes (bias/variance tradeoff)

25. How can we distinguish between natural biological variability and “errors” in the registration? • Repeatedly resample set of imaged embryos to generate a population of virtual pointclouds • Measurements of the relative spatial pattern of any pair of genes in the virtual population should have the same statistics as if we had imaged them directly.

26. Conclusion • Making good use of quantitative data demands correspondence! • correspondences will be published with dataset • Algorithms for computing correspondence: • between embryos at the same stage using common reference gene pattern • between embryos of different stages using average density • Developing methodology for evaluating registration algorithms

28. Local expression descriptor