### Stefan Sommer

Professor at Department of Computer Science, University of Copenhagen

Center for Computational Evolutionary Morphometrics

Stefan Sommer, Rasmus Nielsen, Christy Hipsley, Mads Nielsen

UCPH Data+

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*

- How to model evolution of shape?
- What is the state-space?
- What probability distribution governs the rules of change?

\( \phi \) 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:

- probabilistic model
- tree pruning for shapes
- stochastic shape matching
- MCMC / variational inference:
- likelihoods
- parameter estimation
- gene/character covariance
- interpolation
- hypothesis testing
- tree inference

*inferring the laws of morphological change*

Brown. motion

Brown. motion

Brown. motion

Brown. motion

branch (independent children)

*incorporate leaf observations \(x_{V_T}\) into probabilistic model:*

\(p(X_t|x_{V_T})\)

Doob’s h-transform

\(h_s(x)=\prod_{t\in\mathrm{ch(s)}}h_{s\to t}(x)\)

conditioned process \(X^*_t\)

approximations \(\tilde{h}\)

guided process \(X^\circ_t\)

*4 cases: Phylogenetics and morphology is foundation for much of bioscience*

Canid evolution and dog domestication

Refine palaeohominid tree

Rasmus Nielsen

GLOBE UCPH, Berkeley

Phylogenetics

Stefan Sommer

DIKU UCPH

Shape analysis

Christy Hipsley

BIO UCPH

Morphology

Mads Nielsen

DIKU UCPH

Image analysis

Anders Jordan

Natural History Museum of Denmark

Tom Gilbert

GLOBE

Frido Welker

GLOBE

Guojie Zhang

UCPH BIO

**Elizabeth Baker, Sofia Stroustrup, Marcus Teller, ****Lili Bao, ****Gefan Yang, Liwei Hu, Michael Severinsen, Chao Zhang,**

**Christine Sarah Andersen **+ more to come

Josefin Stiller

UCPH BIO

By Stefan Sommer

- 218

Professor at Department of Computer Science, University of Copenhagen