Computing correspondences in order to study spatial and temporal

patterns of gene expression

Charless Fowlkes

UC Berkeley, Computer Science

1.Why do we need correspondence?2.How can we identify corresponding nuclei in two

different embryos?3.What does it mean for a correspondence to be

correct?

Why do we need correspondence?

Why do we need correspondence?

Correspondence allows us compare the expression of a given gene across different embryos of the same stage.

Why do we need correspondence?

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?

Correspondence allows us to compare gene expression patterns between embryos at different time points

How can we find corresponding nuclei?

How do we find corresponding nuclei?

• Traditional approach: nuclei at the same percent egg length correspond.

• Coarse Alignment:1. align the principal axis of each embryo

2. scale to the same egg length

3. select the appropriate d/v orientation

Nuclei with the same a/p and d/v coordinates correspond

FTZ average aftercoarse alignment

Detailed correspondence based on morphology alone appears quite difficult

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.

Xij = 1 if point i is matched to point j 0 otherwise

Correspondence as optimization

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 : Σi Xij = 1

Σj Xij = 1

λ sets the relative importance of distance versus expression similarity match

j

i

1. Find correspondence by optimizing Xij

2. Smoothly warp source embryo to bring into alignment with corresponding points

3. Repeat…

Problem: correspondence may not be smooth

Solution: iteratively correspond and warp

FTZ average aftercoarse alignment

FTZ average afterdetailed correspondence

X1

X2

Push expression levels forward thru correspondence function X

Building a composite expression map

How about correspondence between different time points?

Idea: use the average shape combined with averagenuclear density to estimate motions

Generating synthetic time-series

1. Place 6000 nuclei in 3D such that they have the average shape and density pattern of early stage 5 blastoderm

2. 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 densitiesOptimizaiton: conjugate gradient

Predicted nuclear movements

What does it mean for a correspondence to be correct?

FTZ average aftercoarse alignment

FTZ average afterdetailed matching

More flexible warping results in sharper reference pattern but poorer prediction about location of other genes (bias/variance tradeoff)

How can we distinguish between natural biological variability and “errors” in the registration?

1. Repeatedly resample set of imaged embryos to generate a population of virtual pointclouds

2. 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.

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

Local expression descriptor