Stochastic Morphometry and
Center for Computational Evolutionary Morphometrics

Stefan Sommer, Rasmus Nielsen, Christy Hipsley, Mads Nielsen

UCPH Data+

Phylogenetics

Darwin 1859

Until ca. 1980: morphological characters

After ca. 1980: DNA

Corbett-Deitig et al. MBE 37: 160-1614

Modeling evolution using Markov chain with state space = {A, C, T, G}.
Transition probabilities of the process is calculated by exponentiating infinitesimal rate matrix.
Likelihood calculated using Felsenstein’s pruning algorithm and then numerically optimized or used in Bayesian inference.
Genomic sequencing provides millions of observations for accurate estimation of trees.

Questions:

Rules of morphological change

Drivers of morphological change (ecology, historical contingency)

Mechanisms of morphological change (genetic basis)

Current state-of-the-art:

Use of landmarks.

Procrustes alignment.

PCA analyses.

Assume top PCs evolve according to a Brownian motion process.

Then use of methods similar to DNA analyses for computation.

Challenges:

Loss of information in the use of landmarks

Loss of information in linearization of nonlinear shape space

Loss of information in only using the top PCs
Lack of biological interpretability and justification.

Lack of modeling flexibility.

State-of-the-art: full shape is reduced to selected landmarks

State-of-the-art: We have high-res digital morphology images but we cannot use the full shape information

\( \phi \) warp of domain \(\Omega\) (2D or 3D space)

landmarks: \(s=(x_1,\ldots,x_n)\) curves: \(s: \mathbb S^1\to\mathbb R^2\)

surfaces: \(s: \mathbb S^2\to\mathbb R^3\)

\( \phi \)

On growth and form, 1917
D'Arcy Thompson

shape \(s_0\)

shape \(s_1\)

stoch. evolution \(s_0\rightarrow s_1\)

dx_t= -\frac12g(x_t)^{kl}\Gamma(x_t)_{kl}dt + \sqrt{g(x_t)^*}dW_t

Riemannian Brownian motion: