Team:OUC-China/Model
Model
Our modeling is divided in two main parts, the basic part and the advanced part:
Xylose pathway
Cellobiose pathway
Adhesion Platform
Xylose Pathway
Cellobiose Pathway
Adhesion Platform
Overview
The aim to the adhesion platform is binding the E,coli to S.cerevisiaes to form a microbial collaboration platform utilizing streptavidin-biotin interaction. Given that we longed to get a better production rate of ethanol using this platform in our project, we made this model to :
1. simulate the kinetic process of adhesion platform, including the coculture-growth of E.coli and S.cerevisiae, the binding-dissociation process of these two organisms, using agent-based modeling (ABM).
2. prove the binding E.coli to S.cerevisiae can increase the production rate of ethanol using ABM;
3. define the Average Normalized Rate Constant (ANRC) to investigate the best experimental conditions for our lab.
Agent-based modeling
Two agent-based models (ABMs) were developed to investigate the adhesion platform, for ABM is a powerful type of model to simulate a complex system in stochastic way, as well as display the real-time animations of the simulated system.
In ABM, we can create thousands of agents and design them by several simple rules that we encoded in them, resemble to that we design the cell to form a device that could achieve some functions in synthetic biology, and set them in some conditions required. Then, we simulate them to see what would happen. This is an impressive method to explore how a complex group system evolves based on individual activity, or as called “emergence phenomenon“, so we use ABM as our major approach in this model.
We use NetLogo to build our ABMs. The source code could be found in our Github.
Binding-Dissociation Model
This agent-based model (ABM) built in NetLogo shows us vivid results of how different types of cells behave in adhesion platform.
The diagram of simulation is as follows:
Figure 1 The diagram of binding-dissociation model.
Hypotheses
To simplify the reality, several hypotheses had been made as follows:
1. Agents are moving randomly obeying Brownian Motion in liquid environment.
2. Binding and dissociation process are considered independent events for SEi(i≤5).
Derivation
Using parameters shown in Table 1, and simplifying the S.cerevisiae cell as a sphere, and E.coli cell as a cylinder, we calculate the solid angle (Ω) of both of them.
\[\Omega=\iint_S{\sin\theta d\theta d\phi}\]Therefore, the solid angle of a S.cerevisiae and an SEi could be given.
\[\Omega_S=4\pi\] \[\Omega_{SE_i}\approx0.11\]The proportion of occupied solid angle could be calculated.
\[\frac{\Omega_{SE_i}}{\Omega_S}=0.96\%\lt1\%\]Although the primary binding process may affect the secondary binding process, the proportion of occupied solid angle is less than 1%, which means assuming binding and dissociation process as independent events is reasonable. Independent events keep the same probability all the time, so that we could calculate the probabilities of both of them.
Binding Probability(Probb)
Given that binding only occurs when a SEi meets with an E.coli cell, the sample space of the binding probability is considered as “A SEi and an E.coli cell meeting each other”.
\[Probb=P("B"\mid"EM") P("EM")\]where “B” equals “Binding” and “EM” equals “Effective meeting”. “Effective meeting”, in other word, is the streptavidin and biotin meeting each other as two cells encounter. Therefore, alternative form if Probb is given.
\[Probb=\alpha \frac{N_{bio} A_{bio}}{A_{S.cere}}\cdot\frac{N_{strep} A_{strep}}{A_{E.co}}\]By bringing parameters in Table 1, Probb is 54.503 % in our model.
Dissociation Probability(Probd)
Not like binding process, which may happens only when two cells encounter, the dissociation process could be taken place at any time, so the Probd is defined as the probability of dissociation in an unit time interval (Δt) following the exponential distribution.
\[Probd=1-{exp} (-k_d \Delta t)\]By bringing parameters in Table 1, Probd is 0.339 % in our model.
*Supplementary: Derivation of Probd function
The dissociation process could be simplified as follows:
And we could write down the ODE function of it.
\[\frac{-d[SE_i]}{dt}=k_d[SE_i]\]By integrating it with \(t:0\to\Delta t\), \([SE_i]:[SE_i]_0\to[SE_i]_0+\Delta[SE_i]\), we have
\[\ln\left(\frac{[SE_i]_0-\Delta[SE_i]}{[SE_i]_0}\right)=-k_d\Delta t\]Alternative form of this equation is as follows.
\[\frac{\Delta[SE_i]}{[SE_i]_0}=1-exp(-k_d\Delta t)\]From the definition of Probd, we could find that
\[Probd=\frac{\Delta[SE_i]}{[SE_i]_0}\]And we give the probability of dissociation.
\[Probd=1-exp(-k_d\Delta t)\]
3. Adhesion platform does not affect the strength of streptavidin-biotin interaction.
Parameters and agents in ABM
| Parameter name | Description | Value | Unit | Sources/Comments |
|---|---|---|---|---|
| SE | Size of E.coli | 2×0.5 | μm | Kubischek HE(Jan 1990) |
| SS | Size(radii) of S.cereivisae | 3.75 | μm | Walker K, Skelton H, Smith K(2002) |
| ΩS | Solid angle of S.cereivisae | 4π | - | Calculated in model |
| ΩSE | Solid angle of SEi | 0.11 | - | Calculated in model |
| Probb | Probability of binding process | 54.503% | - | Calculated in model |
| Probd | Probability of dissociation process | 0.339% | - | Calculated in model |
| Nbio | Number of biotins displayed in S.cereivisae | 16000 | - | Parthasarathy, R., Bajaj, J., & Boder,E. T.(2005) |
| Nstrep | Number of sreptavidin displayed in E.coli | 160000 | - | Park, M., Jose, J., Thömmes, S., Kim, J. I., Kang, M. J., & Pyun, J. C. (2011) |
| Abio | Influential area per biotin | 2.04×10-3 | μm2 | PDB：4WF2 |
| Astrep | Influential area per streptavidin | 3.91×10-5 | μm2 | Daniel, D. M., Drake, E. J., Hong, L. K., Gulick, A. M., & Sheldon, P. (2013) |
| AS.cere | Surface area of a S.cerevisiae cell | 176.72 | μm2 | Calculated in model |
| AE.co | Surface area of a E.cili cell | 6.29 | μm2 | Calculated in model |
| α | P("B"|"EM") | 1 | - | *Assumed |
| Δt | Unit time interval in simulation | 1 | s | - |
| kd | Dissociation rate constant | 3.4×10-3 | s-1 | Wu, S. C., Ng, K. S., & Wong, S. L. (2009) |
| iniGal | Initial concentration of galactose | mM | Experimental data | |
| iniS.cere | Initial amount of S.cerevisiae in simulation | 100 | - | Experimental data |
| iniE.coli | Initial amount of E.coli in simulation | - | Variable: set to different values in the model | |
| Emove | Average moving rate of E.coli cells | - | Variable: set to different values | |
| Smove | Average moving rate of S.cerevisiae cells | - | Variable: set to different values | |
| YE/g | E.coli-biomass yield | /mM | Estimated from experimental data | |
| YS/g | S.cerevisiae-biomass yield | /mM | Estimated from experimental data |
Table 1 Parameters used in ABM
*Assumption made due to the high affinity of streptavidin biotin interaction.
| Agent name | Description | Comments |
|---|---|---|
| E.coli | E.coli cell | - |
| SEi | S.cerevisiae cell combined with i E.coli cells | i=(1,2,3,4,5) |
| galactose | galactose particle | - |
Table 2 Agents applied in ABM
Simulations
We focus mainly on two conditions to analysis our platform: the ratio of initial amount of S.cerevisiae cells and E.coli cells (S:E), and average moving rate of cells.
The ratio of initial amount of S.cerevisiae cells and E.coli cells (S:E)
We change this ratio from 1:10 to 10:1 by changing initial amount of E.coli cells while keeping the initial amount of S.cerevisiae cells as a constant.
Here shows some of the simulation results:
Figure 2. Binding ratio of each type of SEi with S:E = 5:1, 1:1, 1:5 separately.
Figure 3. Percentage of all SEi in adhesion platform with S:E from 10:1 to 1:10 (color form yellow to red).
The average moving rate of cells
We change the average moving rate of cells from 1 μm/s to 10 μm/s with 1 μm/s step and do simulations with S:E ratio is 1:1..
Figure 4. Binding ratio of each type of SEi with moving rate = 1, 5, 10.
Figure 5. Percentage of all SEi in adhesion platform with moving rate = 1, 4, 8, 10 μm/s.
Animations
As agent-based model could directly display the real-time behaviors of adhesion platform, animated simulations are given.
Note that the cycle-like agents with many colors are S.cerevisiae cells. The blue ones are SE0, and those colors from pink to deep red are SE1 to SE5. The purple rod-like agents are E.coli cells, which are very tiny compared to S.cerevisiae cells. Galactose particles are not shown.
Animation 1. Animated simulation of S:E = 1 : 5 in 1000 min.
Animation 2. Animated simulation of S:E = 1 : 1 in 1000 min.
Animation 3. Animated simulation of S:E = 5 : 1 in 1000 min.
Production properties of SEi
Average Normalized Rate Constant (ANRC)
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