Adaptation in a serial-dilution co-culture

A consumer-resource modeling framework in a serial-dilution setup, where $m$ species compete for $p$ nutrients in a series of batches. At each batch, a nutrient mixture is supplied with a fixed composition $\lbrace c_i(t=0)\rbrace_{i=1}^p$ and total amount $c_0=\sum_i c_i(0)$. Similarly, a mixture of species $\lbrace\rho_\sigma(t=0)\rbrace_{\sigma=1}^m$ is added, with a fixed total amount $\rho_0=\sum_\sigma \rho_\sigma(0)$. Species grow until the depletion of nutrients, the culture is diluted, and a new batch starts, while keeping relative species populations. Each species has a metbolic strategy, $\vec{\alpha}_ \sigma$, which is a vector of enzyme levels, while given a fixed enzyme budget $E_\sigma=\sum_i \alpha_{\sigma,i}$, leading to a metabolic trade-off. Nutrient consumption rates are given by $j_{\sigma,i}=\frac{c_i}{K+c_i}\alpha_{\sigma,i}$.

Dynamics within each batch are given by ¹:

$$\frac{d\rho_\sigma}{dt}=\rho_\sigma \sum_{i=1}^p j_{\sigma,i} \quad,$$ $$\frac{dc_i}{dt}=-\sum_{\sigma=1}^m \rho_\sigma j_{\sigma,i} \quad.$$

The main feature of this framework is the inclusion of species adaptation to changing nutrient levels throughout the batch, by allowing the dynamics of the metabolic strategies, given by ²:

$$\frac{d\alpha_{\sigma,i}}{dt}=(\mathbb{P}_{\sigma,i} E_\sigma-\alpha_{\sigma,i})\sum_{i'=1}^p j_{\sigma,i'} \quad,$$

where $\mathbb{P}_{\sigma,i}$ is an indicator function which is 1 whenever species $\sigma$ produces enzyme $i$, and 0 otherwise. The dynamics of this indicator determine the nature of the metabolic adaptation. Note that an adaptor population can only produce a single enzyme type at a time.

The framework includes a few adaptation models (i.e., different dynamical schemes for $\mathbb{P}_{\sigma,i}$; see app__simulations.m) that operate under these guidelines, and is receptible to the addition of different models that may work in this context. Note that this adaptation feature is based on the 2-nutrient case and thus is limited to $p=2$ (however, $p=1$ and $p>2$ dynmics can be simulated with no adaptation).

The dynamics are numerically solved for using MATLAB's built-in ode89 Runge-Kutta solver with adaptive step size. Simulations can be applied in a specific, instant manner, or by parallelly executing and collecting data from large sets of simulations, using the Split-Apply-Combine method (see below for protocol).

Script Index

Note

Here, the general structure of this repository's Code section and workflow are described. For a more specific script description, look at each script specifically.

• odefun.m and the eventfun*.m functions (one for each adaptation model) are the most basic functions, used by ode89 as function handles to solve for the dynamics. odefun.m contains the actual dynamics, while the eventfun*.m functions are used to track events througout the dynamics.

• The sim__*.m functions carry out the actual simulations (for simulation types, see app__simulations.m). sim__batch.m is the most basic of these, which solves for the dynamics within each batch, and is executed by the other, higher sim__*.m functions (except for sim__invasibility_map.m which uses it indirectly). All functions may optionally plot the results, and (except for sim__batch.m) save the data. The high-end functions sim__serial__interbatch.m, sim__serial__full_dynamics.m, and sim__invasibility_map.m are executed using the apply scripts (see below).

• The app__*.m scripts are the main scripts that apply the simulations, and are directly used by the user for setting parameter values and executing the simulations. app__simulations.m is used for running specific simulations (single or several), whereas the app__slurm__*.cmd files are used for implementing large simulation sets in parallel (see below).

• The plot__*.m functions are used to plot the dynamics or other results (raw or collected data). Some are 'raw' plotters, which are used directly by the sim__*.m functions to plot the results from a single simulation, and have a name corresponding to their simulation function. Others may collect the data from large sets of simulations, either by reading raw-data files or a collected-data table (in the case of inter-batch simulations).

Split-Apply-Combine

A common approach to data-analysis, aiming to implement large simulation sets in an orderly and effective way:

1. Split - Produce a parameter table, such that each row contains values for all moving parameters (that vary between different simulations) for a certain simulation in the set, and a template-file to plug simulation parameters into.
   This is done here by the split_runs__*.m functions.

2. Apply - Perform all jobs using a script that allocates resources and executes them independently in parallel. Each job uses the parameter template to plug values from a certain row in the split table, and performs a simulation. All set-simulations are directed to save raw data in the same directory.
   This is done here by the app__slurm__*.cmd files.

3. Combine - Collect data into a table synchronously from all saved raw-data files in the simulation set. Analyze/plot various dependencies.
   Collecting is done here either by using collect_data__interbatch.m in the case of inter-batch simulations, or directy by some of the plot__*.m functions in other cases.

The power of this method is in its modular nature – jobs can be applied independently; data from existing files can be collected independently at any point in time, regardless of currently running jobs.

Protocol:

$\quad\quad\quad$ split_runs__*.m $\quad \longmapsto \quad$ app__slurm__*.cmd $\quad \longmapsto \quad$ collect_data__interbatch.m* $\quad \longmapsto \quad$ plot__*.m

* If needed

Data and Plots

Data and corresponding figures are automatically saved with easily identifiable, corresponding file names (including the simulation type and important parameter values), in the Data and Plots directories, respectively. Their general structure is embedded and a generic sample of data and figures is included here inside both directories.

Here are some of these figures:

1 Adaptor - Metabolic Strategy distributions (Tip: to simulate a single species, use 2 identical species)

Adaptor VS Non-Adaptor - Invasibility Character

2 Adaptors - Steady-State Populaion Bias VS Sensing Tolerances

References

Footnotes

1. Amir Erez, Jaime G. Lopez, Benjamin G. Weiner, Yigal Meir, and Ned S. Wingreen. Nutrient levels and trade-offs control diversity in a serial dilution ecosystem. eLife, September 2020 (Go to paper). ↩

2. Amir Erez, Jaime G. Lopez, Yigal Meir, and Ned S. Wingreen. Enzyme regulation and mutation in a model serial-dilution ecosystem. Physical Review E, October 2021 (Go to papar). ↩