Team:UCL/Model4
LEGIT - From light to structure
Light Expressed Gene Induced Transcription - A mathematical model that optimizes the speed and precision of 3D structure building with E.coli cells for Barchitecture
Model Aims
1. Determine the rate limiting cellular mechanism
2. Optimize the rate limiting parameters by using a conventional (Sensitivity Analysis) and an innovative (Parameter Sampling) approach
3. Determine the range of optimal light intensities and light pulses our Wet Lab should test
4. Conduct a Cost Analysis to evaluate the economic viability of our optogenetic tool
The Science behind LEGIT
- LEGIT is a mathematical model that determines the conditions required for optimal and rapid
light activated expression of the protein intimin. Intimin is a protein expressed on the surface of our genetically
engineered E. coli cells. We use intimin as an anchor onto which we fuse our binding partners, SpyTag and SpyCatcher.
Once the SpyTags and SpyCatchers have attached to a particular cell they start to bind to SpyTags and SpyCatchers
on other neighbouring cells. We control this process using light. This intracellular binding allows our cells to aggregate
into shapes. In order to control the amount of binding that takes place between our cells we need to control the
amount of intimin expressed on the surface of our cells. LEGIT enables us to define the optimal conditions that should be
used by our Wet Lab in order to increase the expression of intimin on the surface of the cells by 16%.
Breakdown of the cellular mechanisms involved in the expression of intimin:
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We decided to divide our model into four main steps:
Step 1: Photoactivation, dimerization and binding of the Transcription Factor (EL222) to the LuxI promoter
Step 2: Transcription
Step 3: Translation
Step 4: Transport of intimin to the surface of the cell
Figure 1: Demonstrates an overview of the modelling steps in our LEGIT model
LEGIT differs from conventional protein expression models as it incorporates two innovative twists. The first is rate kinetics that are dependent on light induced systems. The second is the transmembranal transport of the intimin protein to the surface of our E.coli cell.
Modelling steps
1) Photoactivation, dimerization and binding of the Transcription Factor (EL222) to the LuxI promoter
Figure 2: Demonstrates an overview of photoactivation
Location: Cytoplasm
The first step of our LEGIT model involves determining the amount of EL222 dimers (EL222D) present in the cell. EL222D are the transcription factors used to activate the expression of intimin in the cell. An EL222 D (EL222 dimer) is produced from the photo-activated dimerization of two EL222 monomers (EL222M) found in the cell. We wanted to establish this relationship because once EL222D binds to the LuxI promotor, it starts to promote the transcription of mRNA.
Since we know that photo-activation is dependent upon light intensity (L) we used the Hill equation in our ODE to evaluate our inducible promoter genetic circuits. We did this so we could evaluate the effect different light intensities would have on the rate of photo-activation of EL222.
2) Transcription
Figure 3: Demonstrates an overview of transcription
Location: Cytoplasm
The second step of our simulated model involves determining the levels of mRNA transcribed from the light induced transcription. Production of mRNA is dependent upon the concentration of EL222D and the rate at which EL222D binds to the LuxI promoter. We took dilution rates due to both cell growth/division and mRNA degradation into consideration when accounting for mRNA equilibriums.
3) Translation
Figure 4: Demonstrates an overview of transcription
Location: Cytoplasm
The next step of our model seeks to establish the production levels of the surface protein intimin from translation. The rate of intimin production is subject to changes in the rate of mRNA translation. Once intimin is translated, it is continuously transported to the surface of the E. coli cell, therefore intimins' concentration in the cell also decreases. To account for fluctuation, we created a function that takes the transport rate of intimin to the surface to the E. Coli cells into consideration as well.
We expect the speed at which intimin is transported to the surface of the cell to decrease over time. This is because as time passes there will be a decrease in available space for intimin on the cell surface. We adapted the Michaelis Menten equation to allow for the consideration of: the intimin proteins already occupying space on the cell surface; (PSurface) and the maximum available space for intimin proteins on the cell surface. The variable was set as the limit for the maximum number of intimin proteins that can be expressed on the cell surface, thus preventing any other proteins from becoming expressed.
Finally, we also decided to take into consideration the fact that dilutions can occur due to cell growth and the half-life of the intimin protein. As such we introduced a degradation term to represent such degradations of intimin over time.
4) Transport
Figure 5: Demonstrates an overview of the transport of intimin to the surface of the cell
Location: Cell surface
Our final differential equation simulates the change in concentration of intimin expressed on the cell
surface over time. This equation takes both the rate at which intimin is being transported to the cell surface
and the intimins’ degradation rate into consideration.
Results
Using our LEGIT model, we determined the optimal range of values our Wet Lab team should seek to experiment with, in order to maximize the expression of intimin on our cells' surface.
Determining the rate limiting step in intimin expression
In order to determine the rate limiting step in the cellular mechanisms involved in intimin
expression we used rate kinetics modelling.
Figure 6: Demonstrates the rate of the cellular mechanisms involved in the expression of intimin (More specifically the rate of: EL222D production and binding to the LuxI promoter; Transcription; Translation; and Surface expression of intimin)
From Figure 6 it is evident that the concentration of the surface expressed intimin appears to be much lower than the other species produced from previous cellular mechanisms. As such we decided to plot a second graph to evaluate only the rate of intimin transportation to the cell surface on a rescaled plot.
Figure 7: Depicts the rescaled Graph to view the rate of Surface Expression of intimin
| Step | Time taken to plateau (hr) |
|---|---|
| Dimerization of EL222 | 1.5 |
| Transcription | 1.7 |
| Translation | 1.9 |
| Transport of intimin to the surface | 2.1 |
Figure 8: Outlines the different steps involved in the cellular expression of intimin and the respective times taken for the concentration of the product species from each step to plateau
A rate limiting step in a process is the step that takes the longest to reach its maximum value- or the one that takes the longest to plateau. The rate limiting step in this cellular mechanism is the transport of intimin to the surface of the cell since, as demonstrated in Figure 8, it takes the longest time to plateau. Because speed is essential to our LIT technology we decided to optimize the rate limiting parameters within the ODE describing intimin transfer to the cell surface.
Optimizing Light intensity using Sensitivity Analysis
Our LEGIT model is initiated with the photoactivation of our EL222M transcription factor. Therefore, we thought it would be interesting to evaluate the effect different light intensities had on the concentration and rate of expression of intimin on the cell surface. The range of values we selected to test fell within the range of 0 W/cm2 and 70 W/cm2, where 0 W/cm2 represented our system in its off state and 70 W/cm2 represented the maximum light intensity E. coli can withstand before they start to die.
Figure 9: Demonstrates the effect light intensity has on the concentration and rate of expression of intimin on the cell surface. It determines the optimum light intensity for our optogenetic tool is 18 W/cm2
From Figure 9 it is evident that the most optimum light intensity for intimin expression is 18 W/cm2. As light intensity increases the heat exerted onto our cells also increases, therefore we expected beyond a certain point (in this case 18 W/cm2) the concentration of intimin expressed on the cell surface to decrease. This could be because the heat exerted from the higher light intensities denatures our cells. Interestingly, at 0 W/cm2 we still experience some intimin expression due to leakage of promoter expression. The concentration of intimin expressed on the cell surface at a light intensity of 0 W/cm2 was assumed to not be sufficient to cause cell adhesion.
Wet Lab contribution
Real life experimental data could be obtained for this model by extracting EL222 monomers from E.coli cells, exposing them to varying intensities of blue light, and running the samples on an SDS page. One could thus evaluate the effect light intensity has on the extent to which dimerization of EL222 monomers occurs. The amount of dimerization taking place could be measured by looking at the distance the EL222 extracts travel down the SDS page, where a shorter distance would be travelled by the EL222 dimers due to their larger molecular weight.
Optimizing the parameters in the rate-limiting step
We decided to optimize the rate-limiting parameters present in the ODE which describes the transport of intimin from the cell cytoplasm to the cell surface.
The two rate-limiting parameters identified were: Vmax and Km. Where Km represents the affinity the surface of the cell has for the intimin protein and Vmax represents the maximum rate at which intimin can be expressed on the surface of the cell. The maximum and minimum values that could be assigned to each parameter were determined from literature. We decided to optimize the values for each parameter using two approaches, a Sensitivity Analysis and Parameter Sampling.
Click here to check out our approaches and assumptions!
Optimizing the Km value with Sensitivity Analysis and Parameter Sampling
Ideally, we would want a low Km value so as to increase the affinity of the cell surface for the intimin. This would decrease the time taken for intimin to be transported to the cell surface.
Figure 10: Evaluates the effect Km has on the rate of intimin expression on the cell surface with the use of a Sensitivity Analysis
Sensitivity Analyses determine the magnitude of the effect particular parameters have on the output of a model. Therefore, we first decided to run a Sensitivity Analysis to determine/ validate that Km has a large contribution to the concentration of intimin expressed on the cell surface. In order to do this a range of Km values, from other similar cellular mechanisms, were inputted into our model. From Figure 10 we determined the optimum Km value to operate our cellular mechanism at is 5E-06 uM. We therefore, decided to conduct Parameter Sampling to ensure the value we select to optimize our Km to can be realistically achieved in our cellular mechanism.
Figure 11: Evaluates the effect Km has on the rate of intimin expression on the cell surface with the use of Parameter Sampling. It determines the optimum Km value for our engineered cells is 3.19e-04 uM
Parameter Sampling allowed us to take a series of factors into consideration when optimizing the value of Km. The values of Km represented in Figure 11 are more realistic and significantly lower than the ones obtained from the Sensitivity Analysis. This is because a series of factors were taken into consideration when creating a normal distribution of the possible Km values for our technology. These factors include: size, origin and operational pH of the cellular mechanism from which the data was sourced. From Figure 11 we determined the optimum Km value that would result in the highest concentration of intimn becoming expressed on the surface of the cell was 3.19e-04 uM.
Interested in the steps and weighting system we used for our Parameter Sampling?
Optimizing the Vmax value with Sensitivity Analysis and Parameter Sampling
We would want to maximize the parameter of Vmax in order to decrease the time it takes for the surface of the cell to fill up with intimin.
Figure 12: Evaluates the effect Vmax has on the rate of intimin expression on the cell surface with the use of a Sensitivity Analysis
The Sensitivity Analysis in Figure 12 demonstrates that Vmax is a large contributor to the final concentration of intimin that can be expressed on the cell surface. Therefore, we decided to progress and carry out a Parameter Analysis in order to optimize our Vmax value.
Figure 13: Evaluates the effect Vmax has on the rate of intimin expression on the cell surface with the use of Parameter Sampling. It determines the optimum Km value for our engineered cells is 6.0E+05 1/s
Parameter Sampling allowed us to take a series of factors into consideration when optimizing the value of Vmax. From Figure 13 we determined the optimum Vmax value to maximize the concentration of intimin expressed on the cell surface is 6.0E+05 1/s.
Curious about the steps Parameter Sampling actually entails?
Determining the effect light pulsing has on intimin expression
Once we determined increasing light intensity causes a proportional increase to the concentration
and rate of expression of intimin on the surface of the cell, we decided to evaluate the effect other light properties would
have on our system. More specifically, we decided to evaluate whether light pulsing had an effect on intimin expression.
This would allow us to see whether continuously activating and deactivating our light induced system could result in the expression
of a larger concentration of intimin on the surface of the cell.
We first ran a control simulation, where we introduced one long pulse of 7 hours of light, in order to
ensure we received a curved response that would simulate the activation and deactivation of our cellular mechanisms
for a defined period of time.
Figure 14: Evaluates the effect a single pulse has on the rate kinetics of the expression of intimin
From Figure 14 it is evident the behavior of all the species produced, after each step involved in the expression
of intimin, is represented in a dumb-bell shaped curve. This is justifiable, as when the light is turned off we
expect the rate of product production to be smaller than the rate of product degradation. Therefore, as time passes
the concentration of each product from each step is expected to decrease.
We then wanted to evaluate the effect pulsing for periods of 4 hours would have on the expression of intimin
on the surface of the cell.
Figure 15: Demonstrates the effect pulsing light, at 4-hour intervals, has on the rate kinetics of the expression of intimin
From Figure 15 we discovered that if we pulse light on the cells for a period of 4 hours three times within 24 hours, we experienced more than a 10-fold increase in the concentration of intimin expressed on the surface of the cells.
Optimized conditions
The biggest increase in the rate of intimin expression on the surface of the cells occurs when the value of Km decreases and the values of Vmax increases. This is something we expected, as the lower the value of Km the higher the affinity our cells’ membranes would have for intimin and therefore the faster the transport of intimin to the cell surface. In addition, the higher the Vmax the faster the rate at which the reaction occurs. Thus, the combination of both of these phenomena results in faster transport of intimin to the surface of the cell.
| Optimized Parameter | Value |
|---|---|
| Km (uM) | 3.19E-04 |
| Vmax (1/s) | 6.0E+05 |
Figure 16: Demonstrates a summary of the optimized conditions for the model
Running the model with optimized parameters
We ran our rate kinetics model with our newly optimized parameters and determined that our optimized parameters produced a 16% increase in the concentration of intimin transported to the surface of the cells. However, the same amount of time was taken for the intimin expression rate to plateau. Therefore, this means that although the concentration of intimin expressed on the surface of the cell was optimized this step prevails as being the rate-limiting step in the network of cellular mechanisms involved.
Figure 17: Comparison of the optimized and unoptimized rate kinetics for the transport of intimin to the cell surface
Cost Analysis
We ran a Cost Analysis to ensure our LIT optogenetic tool was an affordable technology current labs could use to replace alternative technologies. We focused on identifying the main operational costs a user would incur, where we determined the most critical ones were: the light intensity they used to activate the E.coli cellular mechanism; and the frequency of light pulsing. We took the costs incurred for both methods into consideration, where we attempted to identify the most cost-efficient operational conditions for our tool.
| Light Intensity (W/cm2) | Cost per hour of utility (£/hr) |
|---|---|
| 70 | 0.014 |
| 53 | 0.011 |
| 35 | 0.007 |
| 18 | 0.0036 |
| 0 | 0 |
Figure 18: Outlines the operating costs incurred for operating the optogenetic tool with a range of light intensities with no light pulsing present
We also decided to analyze the impact light pulsing would have on the overall process costs.
| Pulsing Duration (hr) | Cost for total pulsing over 2 hour operation period (£) |
|---|---|
| 2 | 7E-03 |
| 1.33 | 4.E-03 |
| 1 | 3.5E-03 |
| 0.5 | 1.8E-03 |
| 0.25 | 8.8E-0.4 |
| 0 | 0 |
Figure 19: Outlines the operating costs incurred for pulsing the light source of the optogenetic tool over a range of frequencies at a light intensity of 40 W/cm2
We then decided to plot both the costs incurred for different light intensities and the
pulsing durations on the same graph to evaluate their synergistic effect.
Figure 20: Demonstrates a comparison of the effects of light pulsing and light intensity on the overall cost of our optogenetic tool. The best cost trade-off was identified when operating our technology is operated at a light intensity of 30 W/cm2 with a 4.5-hour pulsing frequency
Figure 20 exhibits that the costs incurred to operate our optogenetic tool increases proportionally to the light intensity and the light pulsing frequency used. We decided to define the optimum operating conditions for our optogenetic tool as those at which both lines intersect. Beyond the point of intersection, the operational costs for the optogenetic tool were too high and below the intersection point the activity of the E. coli cell adhesion mechanism would be too slow. As one of the biggest selling points for our technology is its rapidity we decided it would not be feasible to operate at such a slow rate.
We identified the best cost trade-off when operating our technology at a light intensity of 30 W/cm2 with a 4.5-hour pulsing frequency.
Alternative applications of our model
Stock market
Our model could be adapted to characterize the volatile nature of publically listed companies’ share prices in the stock market. One could divide the differential equations into the main factors which affect the share price of a company. These factors could be: the Lombard Rate, which is set by the European Central Bank; the desirability of a company’s shares (set as a function of the market share a company has); and a company’s share price. A decrease in the Lombard rate would result in an increase in the share price of a company, which would lead to an increase in the number of the company’s shares bought from the stock market resulting in an increase of market share the company was gaining in its industrial sector. Through our model financial institutions could identify the time it takes for a change in the Lombard Rate to result in a change in the company’s market share. Simultaneously, once the bottleneck is identified, the company’s management team could optimize parameters (such as advertising or product diversification) to minimize the time taken for this change to occur.
Collaboration with iGEM SVCE 2017
Throughout the summer we had a bilateral collaboration with the iGEM SVCE 2017 team from India. Our collaboration started off with us sending each other useful sources and tutorials on how to approach modelling for cellular mechanisms, as neither team considered itself an expert in the field. Once we both started forming the differential equations for our respective models, we sent each other documents with a summary of our ODEs and provided feedback to each other. Both teams provided constructive feedback on how to present complex information in a simple manner to make it easy to understand for non-modellers, while at the same time we helped refine each other’s differential equations.
LEGIT Assumptions
- Mass action kinetics
- Michaelis Menten
- All reactions are taking place in cells that are plated in a petri dish
- Light intensity is the only factor affecting the intimin protein expression in the cells
- Every cell is activated by the same light intensity, regardless of its position in the petri dish
- There is no light reflection or refraction from the cells
- All cells are at their exponential growth phase
- All cells are equally as probable to express SpyTag and SpyCatcher
- SpyTag and SpyCatcher are constitutively expressed with intimin
LEGIT Species
LEGIT Parameters
LEGIT Initial Conditions
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