Team:OUC-China/Model
Model
Xylose Pathway
Cellobiose Pathway
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 3.1 The diagram of binding-dissociation model.
Assumptions
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 of Hypotheses 2 and Calculations of each Probability
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 |
| P | Probability of biotin: streptavidin complex dissociation process | 0.339% | - | Calculated in model |
| Probd | Probability of SEi dissociation process |
i=1 0.339% i=2 0.677% i=3 1.011% i=4 1.344% i=5 1.674% | - | 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 |
*Assumption made due to the high affinity of streptavidin biotin interaction.
Table 1 Parameters used in ABM
| 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 initial concentrations 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 total amount of cells as a constant.
Here shows some of the simulation results:
Figure 3.2 Binding ratio of each type in the simulation when initial ratio of S＆E to be 1:1.
Figure 3.3 Percentage of all SEi in the simulation when initial ratio of S＆E from 1:10 to 10:1.
The total concentration of cells
We change the initial total concentration of cells from 100 /world to 1000 /world while keeping the ratio of S.cerevisiae and E.coli to be 1:1. There are 10 different initial conditions and each condition was simulated 20 times.
Figure 3.4 Binding ratio of each type in the simulation when total amount of cell to be 500.
Figure 3.5 Percentage of all SEi in the simulation with total amount of cell from 100 to 1000.
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 3.1 Animated simulation of S:E = 1 : 5 in 1000 min.（File: Binding-Dissociation Model Animation S1E5.gif）
Animation 3.2 Animated simulation of S:E = 1 : 1 in 1000 min.（File: Binding-Dissociation Model Animation S1E1.gif）
Animation 3.3 Animated simulation of S:E = 5 : 1 in 1000 min.（File: Binding-Dissociation Model Animation S5E1.gif）
Production properties of SEi
To theoretically explore the properties of ethanol-production rate of each SEi (a single S.cerevisiae cell binding with i E.coli cells), we made some hypotheses to simplify our problem.
Hypotheses
1. The number of transporters of xylose is finite, stable.
2. The transporters of S.cerevisiae remain the their maximum transportation rate.
3. E.coli and S.cerevisiae are act as the “giver” and “receiver” without any other influence on each other.
Parameters and agents of the ABM
| Parameter name | Description | Value | Unit | Sources/Comments |
|---|---|---|---|---|
| \(Y_{e/X}\) | Ethanol yield on xylose | 48 | mM/Mm | Estimated from experimental data |
| \(\mu^{trsp}_{max}\) | Maximum uptake rate of transporter | 0.16 | mM/min | Estimated from experimental data |
| \(S_E\) | Size of E.coli | 2×0.5 | μm | Kubitschek HE (Jan 1990) |
| \(S_S\) | Size (radii) of S.cereivisae | 3.75 | μm | Walker K, Skelton H, Smith K (2002) |
| \(R_{diff}\) | Xylose and ethanol particles diffusion rate | - | pixcor/min | Set to different values in the model |
| \(R_{xp}\) | Xylose production rate | 0.16 | mM/min | *Assumed |
| \(X_{bg}\) | Background concentration of xylose | 81.72 | mM | Estimated from experimental data |
*An assumption made while lacking literature values and experimental values. We assumed this value to be consistent with \(\mu^{trsp}_{max}\) for simplification, and later found that this assumption will not influence the conclusions gained form the simulation results.
| 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) |
| Xylose | Xylose particle | 1 particle=2E-3 Mm |
| Ethanol | Ethanol particle | 1 particle=2E-3 Mm |
Simulations
The simulations were set by these rules:
1. Xylose particles produced by E.coli cells and ethanol particles produced by S.cerevisiae cell are diffused by randomly moving —— brownian motion.
2. The simulation world was set closed in case that the agents would cycle through the world in NetLogo.
3. Each simulation was set to run 500 min for 20 times.
We set diffusion rate to be 0.1, 0.2, 0.3, 0.4 pixcor/s to simulate our model in separate conditions of SE0, SE1, SE2, SE3, SE4, SE5. Here shows some of the simulations.
Figure 3.6 Ethanol producing simulations of SEi (i=0, 1, 2, 3, 4, 5) in 500 min.
We normalized the ethanol production rate as pi, and the ethanol production rate of SEi were shown (i= 0, 1, 2, 3, 4, 5).
Figure 3.7 Normalized ethanol production rate (pi) in diffusion rate of 0.1 pixcor/min.
Figure 3.8 Normalized ethanol production rate (pi) in diffusion rate of 0.4 pixcor/min.
The results demonstrate that binding E.coli to S.cerevisiae could theoretically sharply increase the production rate for single cell, and we think there are two reasons for its effectiveness. The first reason is that the distance between the “giver” and “receiver” was sharply shorten so that the diffusion of xylose wouldn't be a limited factor anymore. The second is that, in micro-environment, as the binding number increases, the xylose produced by surrounding E.coli cells also increase, which offers S.cerevisiae cells the higher concentration of resource.
Animations
Two videos of simulations. One is from SE0, and another is from SE5:
Animation 3.4 Animated simulation of SE1 in 500 min.（File: Diffusion Model Animation SE1 Ddiff0.2.gif）
Animation 3.5 Animated simulation of SE5 in 500 min. （File: Diffusion Model Animation SE5 Ddiff0.2.gif）
Average Normalized Rate Constant (ANRC)
Introduction
The effectiveness of adhesion platform is not consistent with binding ratio of SEn, which means a high binding ratio may not lead to high production rate of ethanol. For example, there are 1000 E.coli cells and 10 S.cerevisiae cells in adhesion platform, so the binding ratio may be very big, but the amount of S.cerevisiae cells are too small to have a high production rate. So the effectiveness of adhesion platform should also consider the relative amount of S.cerevisiae cells.
And this effectiveness is not just consistant with the production property of a single SEi. It’s multiple interactions from a group of cells with different binding types, different binding ratios, different production properties. In other words, this is a complex system to be investigated its effectiveness.
So we define an equation named Average Normalized Rate Constant (ANRC) to analyze our simulation results, and give us a hint which experimental conditions could lead to the best production.
\[ANRC=\frac{S}{S+E}\sum^n_{i=0}(p_i\times a_i)\]S —— Number of S.cerevisiae cells. E —— Number of E.coli cells. pi —— Normalized production rate constant from ri of each SEi. αi —— Proportion of SEi. \(\alpha_i=\frac{SE_i}{S}\times 100\%\)
And the difference between adhesion platform and non-adhesion platform could be defined as their difference of each ANRC (Δ), which would tells us whether our system better than non-adhesion platform.
\[\Delta=ANRC_{adhe}-ANRC_{non-adhe}\]ANRC and Δ contained two components. One is the results gained from “Binding-Dissociation Model”, which are S, E, αi, another is the results get from “Production Properties of SEi”, which are values of pi. So this function is the combination analysis of both binding-dissociation and ethanol production.
Supplementary: Derivation of ANRC and Δ
Derivation of ANRC
By adopting the Monod function and stoichiometry rules in our adhesion platform, the production could be simplified as:
The production rate of ethanol is given.
\[V_{eth}=\frac{d[ethanol]}{dt}=\sum^n_{i=0}SE_i\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]The alternative form of this function is:
\[V_{eth}=S\times\sum^n_{i=0}\alpha_i\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]If we consider \(v_{eth}\) as the addition of i \(v_{ethi}\), we would get
\[V_{eth_i}=S\alpha_i\times\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]We could find that the production rate constant of \(v_{ethi}\) is the latter item in function, which we defined as \(r_i\).
\[r_i=\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]This means the factors determining the rate constant of each type of SEi are (1) \(k_i\),\(K_i\) the characteristics of SEi which are the same for all type of SEi, because their ethanol production unit are S.cerevisiae cell. (2) [xylose] the partial concentration of the xylose surrounding each cell, and this is the difference between different types of SEi, because the partial concentrations of SEi is higher than that of SEi 1.
The partial concentration of xylose for SEi could be divided into two parts —— a part produced from E.coli cells binding it ([xylose]self), and a part from other E.coli cells as background ([xylose]bg).
\[[xyloe]=[xylose]_{self}+[xyloe]_{bg}\][xylose]bg is the stable concentration of xylose in this system, and is determined by its production rate and consumption rate. Easy to notice that if [xylose]bg is high enough, the concentration of xylose ([xylose]) would be nothing different between adhesion platform and non-adhesion platform, which means the advantage of adhesion platform would not be so significant. Therefore, to overcome it, one possible solution is to accelerate the process of consuming xylose by increasing the transportation rate of the transporters on S.cerevisiae cells to decrease the [xylose]bg. It seems that the transporter is the key factor of making adhesion platform significant.
[xylose]self is the characteristic property of each type of SEi that could not be affected by environments. Easy to find that simulations in “Production Properties of SEi” are to get ri values by setting the amount of S.cerevisiae cells to be 1.
By normalizing the ri values to the maximum of them, we have the normalized production rate constant pi.
\[p_i=\frac{r_i}{max(r_i)}\]So the normalized ethanol production rate is as follows:
\[[Normalized_{V_{eth}}]=S\times\sum^n_{i=0}(a_i\times p_i)\]To eliminate the distinctions resulting from different amounts of cells in different systems, we take the average of the [Normalized veth] to obtain the mean level of ethanol production rate per cell, and defined it as Average Normalized Rate Constant (ANRC).
And the equation of ANRC is also given:
\[ANRC=\frac{S}{S+E}\sum^n_{i=0}(p_i\times a_i)\]Derivation of Δ
Wondering whether our adhesion platform would have a better performance than non-adhesion platform, we defined Δ.
\[\Delta=ANRC_{adhe}-ANRC_{non-adhe}\]From the definition of ANRC, easy to get \(ANRC_{non-adhe}\) .
\[ANRC_{non-adhe}=p_0\times\frac{S}{S+E}\]Then the function of Δ is as follows:
\[\Delta=\frac{S}{S+E}\sum^n_{i=0}[(p_i-p_0)\times a_i]\]Analysis of adhesion platform
By changing the conditions in “Binding-Dissociation Model”, we calculated ANRC and Δ of each simulations. Results shown as follows.
Figure 3.9 ANRC values of different initial ratios of S.cerevisiae and E.coli.
Figure 3.10 Δ values of different initial ratios of S.cerevisiae and E.coli.
Figure 11. ANRC values of different initial concentrations of cells.
Figure 12. Δ values of different initial concentrations of cells.
Given that Δ values are bigger than zero in our simulations, we now can conclude that our adhesion platform would get a better performance in ethanol production than non-adhesion platform.
By analysis above, we can conclude that the best initial ratio of S.cerevisiae and E.coli is XX : XX, and initial concentration of cells is XX. This conclusion is mirrored to the wet lab, and is meaningful for optimizing our project.
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