Team:UCL/Model1
Model Aims
1. Visually represent cell adhesion
2. Combine gene expression kinetics and cellular automata for spatial analysis
3. Incorporate Artificial Intelligence with a Genetic Algorithm to automate parameter optimization
4. Input experimental data from the Wet Lab for parameter optimization
5. Determine the optimum initial light pattern to activate the cellular mechanism
6. Determine the optimum light intensity for cell adhesion
The science behind GOLIT
GOLIT is a model that creates a visual representation of how solid cell structures can be formed by inducing the expression and binding of SpyTag and SpyCatcher, which are expressed on the surface of our E.coli cells and are explored in our Barchitecture section. GOLIT is inspired from Conway’s Game of Life.
We are modelling a cellular automaton to dynamically represent the adhesion of cells.
We started by collaborating with the Wet Lab team to engineer E. coli cells to express the surface protein intimin.
Then we got the binding partners SpyTag and SpyCatcher to attach to the surface protein (this expression system is modelled in the LEGIT model ).
When SpyTag and SpyCatcher from neighboring cells are in close proximity to each other they form a covalent bond. This allows cells to aggregate and form bacterial structures.
With GOLIT we are simulating how the concentration of SpyTag/SpyCatcher, expressed on the surface of the cells, influences the rate at which cells adhere to each other. We obtained the different concentrations for SpyTag/SpyCatcher from our LEGIT model, where different concentrations of SpyTag/SpyCatcher were expressed depending on the light intensity used to photoactivate the cellular mechanism.
How did we adapt the Game of Life?
Similar to Conway's Game of Life, we are representing the cells on a 2D grid so when running the model, the cells are constantly updated over a defined period of time. In the GOLIT model the cells can either be in the “bound” or “unbound” state. Our game requires the user to define the state of all the cells on the grid and update the grid according to a set of rules for each time point. The state of a cell at a particular time point depends on the state of its neighboring cells as well as the concentration of SpyTag/ SpyCatcher expressed on the surface of the target cell.
The rules below assign probabilities of one cell becoming "bound" depending on the number of 'bound' neighbors surrounding it.
Rules of the Game of LIT
| Number of "bound" neighbors | Probability of cell changing to bound state (%) |
|---|---|
| 0 | 20 |
| 1 | 30 |
| 2 | 60 |
| 3 | 80 |
| 4 | 99 |
Figure 1: Demonstrates the rules of the game, where the probability of cell adhesion depends on the number of bound neighbours a cell has
Combination of ODE and Cellular Automata
Aim: Determine which light intensity allows for the fastest cell aggregation, i.e. binding of SpyTag and SpyCatcher
The particularity of the GOLIT model lies in fact that we are combining rate kinetics of protein expression with cellular
automata for spatial analysis. The concentration of SpyTag and SpyCatcher expressed on the surface of the engineered E. coli
cells is critical for the efficiency of cell adhesion. The LEGIT model uses rate kinetics to model the rate of intimin expressed
on the surface of the bacteria for light intensities within the range 0-70 W/cm2.
For the purpose of this model we are assuming that the concentration of intimin expressed corresponds to the amount of SpyTag/ SpyCatcher on the surface of the cells. Since updating the grid in the GOLIT model depends on the concentration of protein expressed, we extracted the concentrations of intimin expressed from the LEGIT model and fed them into the GOLIT model.
- The initial conditions that need to be set up at the beginning of each game are:
- Dimensions of the 2D grid (width and length)
- The initial amount of bound cells at zero time point, t=0
- The initial amount of unbound cells at zero time point, t=0
- The number of generations, i.e. the number of times the grid will be updated
- The duration of each generation
Click here to check out our code on Github!
Results
We ran the GOLIT model for light intensities 0W/cm2, 18W/cm2, 35W/cm2, 53W/cm2, and 70W/cm2, i.e. the same values tested in the intimin expression model (LEGIT). For each light intensity tested, we plotted the number of cells vs. time points in order to determine the maximum amount of adhesion that was accomplished/feasible for each condition evaluated.
- Setting up the conditions for the game:
- Dimensions of the 2D grid: 50x50 matrix
- Evaluate the adhesion of cells for 15 generations
- Initially at time, t=0, no cells are in the "bound" state
- Black cells represent "bound" cells and white cells are "unbound" cells
Figure 2: For light intensity 0 W/cm2, the plot shows the number of bound cells achieved over our pre-defined time-frame, 15 generations. The grid visually represents the amount of bound cells after 15 generations
Figure 3: For light intensity 18 W/cm2, the plot shows the number of bound cells achieved over our pre-defined time-frame, 15 generations. The grid visually represents the amount of bound cells after 15 generations
Figure 4: For light intensity 35 W/cm2, the plot shows the number of bound cells achieved over our pre-defined time-frame, 15 generations. The grid visually represents the amount of bound cells after 15 generations
Figure 5: For light intensity 53 W/cm2, the plot shows the number of bound cells achieved over our pre-defined time-frame, 15 generations. The grid visually represents the amount of bound cells after 15 generations
Figure 6: For light intensity 70 W/cm2, the plot shows the number of bound cells achieved over our pre-defined time-frame, 15 generations. The grid visually represents the amount of bound cells after 15 generations
Note that the GOLIT model is based on probabilities and that exact values will change every time the model is run. We are only interested in the general trends rather than precise numbers.
| Light intensity (W/cm2) | Number of "bound" cells | Percentage of bound cells (%) |
|---|---|---|
| 0 | 264 | 11 |
| 18 | 2014 | 81 |
| 35 | 2008 | 83 |
| 53 | 1684 | 67 |
| 70 | 1640 | 66 |
Figure 7: Demonstrates a summary of the efficiency of binding for each light intensity tested
Given that the dimensions of the grid were 50 columns x 50 rows, we know that the maximum number of "bound" cells that can exist on the grid is 2500. The ultimate aim of the GOLIT model is to obtain a black grid, i.e. 100% bound cells. To start off we ran the model for 15 generations and compared the performance of each light intensity tested. From the LEGIT model we know that 18W/cm2 gives the highest production of surface protein. We observe a similar result when running the GOLIT model, since 18W/cm2 gives the highest number of bound cells after 15 generations.
When we ran the model for 15 generations, focus was not put on filling up the grid but rather on comparing the rates at which the grids fill up, i.e. rates at which the cells aggregate. Now, given that 18W/cm2 represents the most efficient conditions to for bacterial cell adhesion, we wanted to test the number of generations that are required to fill up the grid.
Figure 8: shows the number of aggregated cells after 80 generations for 18W/cm2, which we identified as the light intensity giving the highest rate of cell aggregation
Making the Model smart: incorporating Artificial Intelligence
Aim: Optimize the initial light distribution pattern on the grid to allow for the fastest rate of cell aggregation.
We decided to upgrade our model by incorporating artificial intelligence, more particularly a
Genetic Algorithm (GA) . Genetic Algorithms are a type of subsymbolic artificial agent. Essentially, they adopt the
principles of Darwinian evolution to tackle the challenge of parameter optimization.
In the context of GOLIT we are applying a genetic algorithm to optimize the positions at which light should be shown onto an empty grid,
i.e. the grid at time t=0. In other words, this means that we are using Artificial Intelligence
to generate the initial pattern of “bound” cells on the grid that will allow for the fastest rate of cell
adhesion. The only input arguments given are the maximum number of cells to be placed on the initial
grid. The model then runs a number of “agents” that generate different patterns according to random
probabilities. The genetic algorithm selects the “best agent”, i.e. the “agent” showing the pattern
that will allow cells to aggregate fastest and therefore allow for the most efficient cell aggregation.
How does it work?
Figure 9: Demonstrates the series of steps the genetic algorithm we created follows in order to identify the best light pattern we should use to activate cell adhesion in our cells
Results
As described above the genetic algorithm is an automated way of optimizing the
conditions of the grid. In our case the dependent variable that we want to optimize is the time it takes
to fill up the grid, i.e. optimize the model such that the lowest possible number of generations is needed to
fill up the grid. The independent
variable that affects time is the initial light pattern shown onto our empty grid.
Once the genetic algorithm has selected the "best agent", it provides two main results: the probabilities
we should set for the rules of the game and the number of generations that will be needed to fill up the
grid with those probabilities.
- When running the model in Python, the genetic Algorithm prints these results:
- Finding Cover time: this indicates the number of generations it will take for the grid to fill up, assuming the probabilities given by the best agent are used
- Finding best agent: the 5 probabilities listed correspond to the "optimal" rules one would need to follow when shining light on the grid in order to allow the cells to aggregate at the fastest rate
Feeding the Model with Experimental Data
The ultimate goal of developing mathematical models is to create robust systems which are representative of reality. The best way to optimize parameters in a model is to use experimental data. Our Wet Lab team simulated the bacterial chemical adhesion between SpyTag and SpyCatcher, by testing the adhesion between Biotin and Avadin. Similar to SpyTag and SpyCatcher, Biotin and Avadin form a covalent bond when in close proximity to each other. This experimental data allowed us to determine the optimum concentration of surface proteins needed to achieve binding at the fastest rate.
The video above shows 8 eppendorf tubes, each representing the different experimental conditions tested. We performed a series of experiments with competent cells. From left to right: DMF solvent and Avadin, DMF solvent and PBS solvent, PBS solvent and Avadin, PBS solvent, 125 µMBiothin and Avadin, 125 µM Biotin and PBS, 1000µM Biotin and Avadin, 1000µM Biotin and PBS.
Figure 10: Shows the number of aggregated cells vs. time. These are the experimental results obtained for the simulation of SpyTag/SpyCatcher adhesion using Biotin and Avadin
From our experimental data we determined that as time passed a greater number of cells adhered to each other. Therefore, as time passed a greater number of cells aggregated to each other, and sedimented to the bottom of the eppendorf tubes, a lower concentration of un-adhered cells remained in the supernatant stream. To implement this data in the GOLIT, we calculated the percentage change in aggregation between each time point. This percentage change represented the probability of a cell adhering with another cell at each time point. We decided to run the Game of LIT once we combined these experimentally determined probabilities with the binding probabilities that were dependent on the number of neighbouring cells.
The grid, as expected, does not fill up at the end our experiment. This occurs because from the final OD measurement of the supernatant stream in the eppendorf tubes it was evident that un-adhered cells were still present in the supernatant stream. We would expect all the cells to adhere to each other if we allowed for a sufficiently long period of time.
We would like to conduct future experiments to determine the optimum concentrations of Avidin and Biotin required to lead to all the cells present in a culture to adhere to themselves in the least amount of time.
Click here to check the protocol of the cell aggregation experiment
Assumptions
- Assume no metabolites are present on the surface or in the solution with the cells, i.e. biotin only attaches to surface protein
- Assume no cell lysis during centrifugation (All cells are whole)
- Assume the initial concentration of cells is (2.5*107 E.coli cells)
- Assume there are only biotinylated cells present
- Assume initial concentration of Avidin is 20 mM
Bibliography
Introduction to Genetic Algorithms [Internet]. Doc.ic.ac.uk. 2017 [cited 8 October 2017]. Available from: https://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/hmw/article1.html
Game of Life -- from Wolfram MathWorld [Internet]. Mathworld.wolfram.com. 2017 [cited 8 October 2017]. Available from: http://mathworld.wolfram.com/GameofLife.html
