Team:Ruperto Carola/MMT

## Modelling evolution Usually, models of cell growth and gene expression assume a fixed genetic background for the cells involved. This does not play well with the idea of evolution, introducing mutation and selecion. Once we add that, standard models break down. To properly model evolution, we need to integrate both the behaviour of our cells' gene regulatory networks and their competition in the context of a larger population. Understanding the model of this at once can be quite the brain-melter, so let us take this step-by-step. We begin by setting down the equations for the population dynamics of a single type of cells, assuming our media can sustain the growth of infinitely many cells. As we're looking at the dynamics of our population \(N : T \to C\) here, we're looking at the rate of change in cell concentration in time, which is made up of a rate of cell division \(\gamma : \mathbb{R}_+\) and a death rate \(\delta : \mathbb{R}_+\). \[ \frac{dN}{dt} = \gamma N - \delta N \] Such media of course do not exist, and realistic conditions dictate a finite carrying capacity \(K : C\). We can easily adjust above equation to enforce a given \(K\) using logistic growth \[ \frac{dN}{dt} = (\gamma - \delta) \cdot N \left( 1 - \frac{N}{K} \right) \] or by explicitly coupling to nutrient concentration \(S : T \to C\) via the Monod equations: \[ \frac{dN}{dt} = \frac{\gamma S}{K + S} N - \delta N \] \[ \frac{dS}{dt} = -\upsilon \frac{\gamma S}{K + S} N \] where \(\upsilon : \mathbb{R}_+\) is a yield factor giving nutrient consumption per unit growth. That's it for the basics. To take the first step towards modelling evolution, consider a set of possible genotypes \(\Gamma : \mathsf{FinSet}\), and the concentration of a subpopulation of a given genotype \(N : \Gamma \to T \to C\), with genotype-specific growth rate \(\gamma : \Gamma \to \mathbb{R}_+\). You can think of differences in \(\gamma\) across \(\Gamma\) as _differences in fitness_ between the genotypes. Thus, subpopulations of fixed genotype compete for nutrients in media: \[ \frac{dN_k}{dt} = \frac{\gamma_k S}{K + S} N_k - \delta N_k \] At this point, we have recovered several aspects of evolving systems – the presence of multiple genotypes, as well as selection _via_ competition of multiple subpopulations for growth and media – but we are still missing one last aspect, without which everything breaks down. We are missing mutation, which provides a way for individuals to move between genotypes \(k : \Gamma\). In most systems, DNA mutation is tightly coupled to DNA replication, which is in turn coupled to cell division. Thus, we want an extension to our model, which moves individuals across genotypes as they divide. We can achieve this by introducing an additional term \(\pi_{k' \to k} : \Gamma \to \Gamma \to \Delta^1\) giving the probability of mutating from genotype \(k'\) to \(k\), at each cell division. Coupled into our model, we arrive at: \[ \frac{dN_k}{dt} = \sum_{k' : \Gamma} \pi_{k' \to k} \cdot \frac{\gamma_{k'} S}{K + S} \cdot N_{k'} - \delta N_k \] where we gather up contributions of mutation from each subpopulation by summing over \(k' : \Gamma\). Now, we have all the parts and we can rewrite this equation into a more succinct, descriptive form: \[ \frac{d\mathbf{N}}{dt} = \gamma_0 \mathbf{\pi}\mathbf{f} \mathbf{N} - \delta \mathbf{N} \] where \(\mathbf{N} : \Gamma \to T \to C\) is the vector of concentrations \(N_k\) for each subpopulation of genotype \(k : \Gamma\), \(\gamma_0 : \mathbb{R}_+\) a maximum growth rate, \(\mathbf{f} : \Gamma \to (0, 1)\) a relative fitness (relative growth rate) and mutation matrix \(\mathbf{\pi} : \Gamma \to \Gamma \to \Delta^1\) with elements \(\pi_{k' \to k}\). Note that for evolution of traits involving interaction between cells, the fitness \(\mathbf{f}\) may have an arbitrary dependence on \(\mathbf{N}\). Now, the observant reader may wonder, where we get our fitness \(\mathbf{f}\) from. Depending on the level of abstraction we want to treat our system at, we may indeed directly specify a function \(f\) from genotypes \(\Gamma\) to fitness values \((0, 1)\). However, we may also specify actual gene regulatory networks in our cells, and work our way up from there. Briefly, the second case involves treating gene expression as a fast process, relative to cell division and assuming partial steady state behaviour. For more information, take a look at [gene regulatory networks](). Our model differs from the majority of the literature on modelling evolution, as we focus on non-saturated liquid-culture with population growth. To see how our model relates to the literature, refer to [game theory](). Having laid the foundations for our model, let us now move on to a specific system for directed evolution in yeast.

Modeling

Mathematical models and computer simulations provide a great way to describe the function and operation of BioBrick Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior in conjunction with experiments in the wetlab.

Gold Medal Criterion #3

Convince the judges that your project's design and/or implementation is based on insight you have gained from modeling. This could be either a new model you develop or the implementation of a model from a previous team. You must thoroughly document your model's contribution to your project on your team's wiki, including assumptions, relevant data, model results, and a clear explanation of your model that anyone can understand.

The model should impact your project design in a meaningful way. Modeling may include, but is not limited to, deterministic, exploratory, molecular dynamic, and stochastic models. Teams may also explore the physical modeling of a single component within a system or utilize mathematical modeling for predicting function of a more complex device.

Best Model Special Prize

To compete for the Best Model prize, please describe your work on this page and also fill out the description on the judging form. Please note you can compete for both the Gold Medal criterion #3 and the Best Model prize with this page.

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Inspiration

Here are a few examples from previous teams: