Mathematics of Development and Evolution
We can consider complex systems to be ones formed by a large number of heavily interacting elements. As a result, many of mankind’s greatest challenges come from trying to unravel the behaviour of these systems, such as the climate, the economy, society, the brain, biological development, etc. However, contrary to this, the hydrogen atom, solar system or an ideal gas would be simple systems, despite the fact that in order to study them we need to use in-depth physics concepts and sophisticated mathematics.
Our group is focused in understanding the mathematical bases of evolution.
The main question we want to address is: how did complex organisms arise in evolution? Or more in general, how can complexity evolve also in other systems like culture, society and molecular pre-biotic systems. In the case of multicellular organisms, such as us, this main question translates into three other questions that we want to address:
1. How does a fertilized egg cell transforms into a complex adult organism? This is a complex functional organism characterized by many cells, cell types and a specific distribution of those in space. Can we understand the mathematics of such a dramatic pattern formation process?
2. How did this complexity arose by a gradual process of evolution by natural selection. This implies explaining also the evolution of the development that produces such complexity in each generation.
3. Are there some logical or mathematical requirements or principles that gene networks need to fulfill in order to be able to produce complex morphologies during development? If so, can we approach question 1 and 2 above from understanding these principles?
Understanding question 1 is highly non-trivial: something very complex, our body, is produced from something much simpler and small, a simple cell, in a relatively short time.
This process can not be understood by looking at single genes. Embryonic development involves the interaction between huge numbers of genes and cells. Thus, for example, to understand which morphological changes will occur from specific mutations in a gene, we need to understand how that gene is embedded in a gene network and how that affects the dynamics of signaling and mechanical interactions between cells and tissues. In other words, we have a huge number of heavily interacting elements at different levels (e.g. genes, cells, tissues) that lead to the arising and variation of a macroscopic pattern, the body’s.
To address question 1 we build multi-scale models of embryonic development. Each such models includes a set of differential equations describing how genes regulate each other’s expression and a set of differential equations describing how cells move, change shape and activate cell behaviors (cell growth, cell contraction, cell division, etc…). Each cell contains the same set of genes and equations, but, as a result of model dynamics, different cells end up expressing genes at different intensities. Genes affect the mechanical properties and behaviors of the cells in which they are expressed. As a result, cells move and rearrange themselves in space and, in their turn, affect back gene expression by differentially affecting, through cell-cell signaling, where genes get expressed. The models are, essentially, reaction-diffusion models in which the shape of the compartment in which diffusion is taking place, the embryo, is changing over time as a consequence of the reactions, i.e. gene regulation, affecting cell movement. Through specific gene networks, initial conditions and regulation of cell behaviors and mechanical properties by gene products, we simulate how the morphology of adult organs develops. In other words, the model reproduces the position in 3D space and the gene expression in each cell in the adult and how that has changed over development. We do that for specific organs in collaboration with experimentalists (e.g. teeth (Salazar-Ciudad and Jernvall, 2002, 2010; Salazar-Ciudad 2008, 2012; Järvinen et al., 2006; Harjunmaa et al., 2014; Renvoise et al., 2017), fly wings (Ray et al., 2015), turtle carapaces (Moustakas-Verho, 2014) among others) but also for the ensemble of animal development (Salazar-Ciudad, 2000, 2001a, 2001b, Marin-Riera et al., 2016).
For question 2 we take two different approaches. In a populational one we combine the models of development we build for question 1 with models of mutation, natural selection and genetic drift in populations. In a way we put together the mathematical apparatus of populational genetics with that of the multiscale models of embryonic development. This way we simulate how complex morphologies arise in evolution. In the ensemble approach we build a huge number of random networks and check which ones are able to produce complex morphologies.
From both approaches we want to understand how phenotypes and development evolve and, also, if there are some logical or mathematical requirements that gene networks need to fulfill in order to be able to produce complex morphologies during development. If that would be the case, as we think it is, the study of development and evolution would be greatly simplified. The evolution of development could then be understood, as we attempt, by looking at the networks fulfilling these requirements and exploring how likely they arise from random mutation, or how likely one can be transformed into another, as compared with the adaptive value of the morphologies they produce (Salazar-Ciudad et al., 2001a, Salazar-Ciudad 2010). This will entitle us with building a general theory of how development works, and most importantly, of how it evolves.
In parallel we also explore how our evolution and development approach can be applied to evolution in culture (Salazar-Ciudad, 2010c) and in the origins of life (Salazar-Ciudad 2008b, 2013b). In these approaches development is just replaced by the processes that lead to phenotypic variation.
Part of our research also involves discussing how the insights acquired from our models modify several aspects of evolutionary theory such as developmental constraints (Salazar-Ciudad, 2006), canalization (Salazar-Ciudad, 2007), robustness (Salazar-Ciudad, 2007), novelty (Salazar-Ciudad, 2006b) or just evolutionary theory in general (Salazar-Ciudad, 2008).
In addition, we also have data-mining approaches in what we call statistical developmental biology. This is the combined analysis of massive transcriptomic and genomic databases to quantitatively test several long-held hypothesis on the evolution of development, this goes from an explicit measurement of how complexity increases over development and anatomy (Salvador-Martinez and Salazar-Ciudad, 2015, 2017) to an estimation of a map of acting selection over the body of the fly (Salvador-Martinez et al., 2018).
We added an image, an example of a developmental mechanism with a gene network combining different cell behaviours implemented in EmbryoMaker. The left diagram show the gene network. The boxes indicate specific cell behaviours or cell mechanical properties regulated by specific genes in the network. a) Initial conditions, hollow spheric epithelium with a single cell (yellow) expressing gene TF1. B) Outcome, after a number of iterations, of the complex developmental mechanism applied on the initial conditions in A. The left column shows, in section, the node types. Blue for basal side of cylinders, violet for the apical side of cylinders, red for mesenchymal cells and orange for extracellular matrix nodes. Middle and right column display concentrations of GF2 and TF5 respectively (yellow for high concentration, blue for low concentration).