From 2014.igem.org

Wageningen UR iGEM 2014

Modeling Overview

The performance of BananaGuard as biological control agent is defined by three factors: its ability to destroy or inhibit Fusarium oxysporum upon detection, its ability to compete with other micro-organisms in the soil and the stability of the genetic circuit.

If the fungal growth inhibitors cannot destroy or inhibit F. oxysporum, the bananaplants will die. This aspect of BananaGuards performance is being tested in vitro. This leaves two remaining questions to be answered:

Can the BananaGuard system survive in the presence of competing micro-organisms?

Is the genetic circuit robust to biological fluctuations?

The performance and stability of the genetic circuit is crucial for BananaGuard to work. To prevent F. oxysporum from functioning, the fungal growth inhibitors in BananaGuard need to be produced when the fungi are present. This requires two aspects of the system to function properly: the toggle-switch within the Kill-switch system needs to be stable and the fusaric acid dependent promoter needs to activate the system in the presence of fusaric acid.

Optimizing promoter design constitutes the first step of BananaGuard development. Different configurations of promoter elements on the promoters of the Kill-switch were modeled using statistical mechanics. Parameters such as repressor binding strengths and repressor production rates were changed in each simulation. By scoring Kill-switch activation, the promoter element configuration with the most simulations showing a bi-stable switch were considered to be most likely to work. Including the anti-gene transfer inhibition effect on cell growth rates and the fusaric acid dependent promoter, this optimized version of the BananaGuard system was subsequently modeled over a population of dividing cells, assessing both its stability and performance in the form of a stochastic model.

Running thousands of simulations incorporating noise terms and promoter leakiness, the robustness of the BananaGuard system was quantified. Stochasticity in the system caused by cell division resulted in fractions of fast and slow growing cell populations with either stable or unstable Kill-switches. The same was considered for system performance; upon induction by fusaric acid a portion of the cell population activated their Kill-switch or remained in the same state. In order to make the simulations as accurate as possible a parameter for the maximum growth rate of BananaGuard was required. This parameter was obtained from a metabolic model that described the effects of competition between the BananaGuard system and micro-organisms in the rhizosphere.

To ascertain our hosts competitiveness the rate of cell division needs to be investigated on a metabolic level. Cell growth rate is a key factor in the competition between our engineered Pseudomonas putida and the wildtype P. putida found in the rhizosphere of the banana plant roots [1,2]. The integrated genetic circuit uses metabolic resources that would otherwise be dedicated to cellular maintenance or growth. Should this divergence of resources be too large our host will be outcompeted. A genome scale metabolic model was extended, investigating and outlining the cost of introducing our synthetic genes on a metabolic level.

Kill-switch promoter design

Introduction

Promoter design for a stable Kill-switch involves balancing of different regulatory components. Two elements, the number of repressor binding sites and the position of those binding sites with respect to the gene, control two key variables: the chance a repressor binds to the regulatory region and the affinity with which it binds. These variables, in turn, determine the effectiveness of the repressor protein [1,2].
It has been shown that repressor proteins are more effective when there are more repressor binding sites in front of a gene, confirming that the increased chance of repressor protein binding leads to a significant increase in its effectiveness [1,2]. With respect to binding affinity, it has been shown for TetR proteins that a repressor binding site positioned in front of the open reading frame is more effective than those positioned at the back [6].
Because of slight variations in the position of a repressor binding site or variations in experimental conditions, a range of different repressor binding strength values are reported for the same repressor (sometimes varying several orders of magnitude)[4,5].
This means that case specific designs cannot be modeled accurately but it is possible to estimate the likelihood that the designed system will work. By taking all the possible repressor binding site combinations and varying the different binding affinities (thereby simulating their position) and production rates, the robustness of the system can determined given the repressor binding site combinations(figure 2). Statistical mechanics is an ideal tool because it can incorporate both repressor binding strength and binding sites. Continue to learn more about the statistical mechanics approach or view the results.

Theory and experimental design

Statistical Mechanics

Statistical mechanics provides a way to understand how the macroscopic properties of a system arise from the individual components within this system. In effect there are two states to consider in statistical mechanic descriptions. A macrostate specifies a system in terms of quantities that average over the microscopic constituents of the system. For example if you consider a closed container of gas (figure 3), quantities for the macrostate would be pressure or volume. This means that these macrostate quantities only work if you consider a system with a very large number of particles. Talking about a macrostate of a single molecule makes no sense since the terms are inherently descriptive of large systems. A microstate, however, does describe the properties of a single molecule e.g. kinetic energy and position. The macro- and micro-states come together in the key concept of statistical mechanics. A lot of different microstates can result in the same macrostate, but characterizing the macrostate imposes restraints on the possible microstates [7].

Modeling Biological Systems using Statistical Mechanics

The logic of macrostates and microstates can also be applied to cells, replacing the container of the previous example with the cell and the different particle constituents with RNA polymerases and repressor proteins. The condition and production states of the various proteins can be considered a macrostate. Modeling with statistical mechanics provides an insight in transcription regulation of a genetic circuit by generating ‘parameter-free’ predictions for the levels of gene expression. Assuming transcription of a gene to be proportional to the chance that a RNA polymerase binds to the promoter of interest a statistical mechanics model has two microstate outcomes [1,2]:

Non-specific binding: A state where none of the polymerases are bound to the gene of interest and distributed among non-specific binding sites .

Specific binding: A state where the promoter of the gene of interest is occupied and the remaining polymerases are distributed among the non-specific binding sites.
In order to determine the probability that either state will be realized we need to know the total number of possible microstates and the chance that a particular microstate will be achieved. For the non-specific binding by RNA polymerases combinatorics can be used [1]. The following equations will describe step for step how statistical mechanics can model biology, starting with combinatorics.

$\frac{N_{n s}!}{P! (N_{n s} - P)!} = N u m b e r o f n o n - s p e c i f i c b i n d i n g p o s s i b i l i t i e s$

Where P is the number of RNA polymerases and N_ns the number of non-specific binding sites. Because there is a difference in the binding strength between specific and non-specific binding sites a Boltzmann weight needs to be included [1]:

$Z (P) = \frac{N_{n s}!}{P! (N_{n s} - P)!} e^{- \frac{P ε_{p d}^{N S}}{K_{b} T}}$

To describes difference between strength of specific and non-specific binding a Boltzmann weight is added. For the non-specific binding of RNA polymerases the Boltzmann weight ε_pd^NS is going to be very low compared to specific binding, compensating for the high number of binding site arrangements. In order to get the total statistical weight the microstate for RNA polymerase bound to the promoter needs to be added [1,2].

$Z_{t o t} (p) = Z (P) + Z (P - 1) e^{- \frac{ε_{p d}^{S}}{K_{b} T}}$

ε_pd^S is the specific binding energy of the RNA polymerase binding to the promoter. To obtain the chance an RNA polymerase is bound we divide the specific binding chance by the total chance [1,2].

$P b o u n d = \frac{Z (P - 1) e^{- \frac{ε_{p d}^{S}}{K_{b} T}}}{Z_{t o t} (p)}$

By dividing top and bottom by Z(P^-1)e^{(-(ε_pd^S)/(K_bT))} the equation takes the form of a Hill function where the assumption was made that the amount of polymerases is a lot smaller than the amount of non-specific binding sites [1,2].

$P b o u n d = \frac{1}{1 + \frac{N_{n s}}{P} e^{- \frac{∆ ε_{p d}^{S}}{K_{b} T}}}$

In this form a fold change of mRNA expression can be correlated to the chance that a polymerase binds (N_ns/P e^{(-(∆ε_pd^S)/(K_bT))}). If the term describing RNA polymerase binding equals zero, the the chance that a RNA polymerase binds is equal to one and there is no change in gene expression. This will most likely be the case in a system without interference. Most biological systems, however, are dependent on regulators, repression is the main method of gene regulation in our designed circuit (see Kill-switch). The effect of repressors can be taken up in previous equation by adding the microstate ‘repressor protein bound’ as a regulatory region to the total statistical weight (see equation below). The term ‘repressor binding’ states that if a repressor is bound a polymerase cannot bind. The equation assumes a high basal level of expression in the absence of repressors. Note that like the previous equation, it takes the form of a Hill function, the gene expression fold change is dependent on the number of repressors [1,2].

$F o l d C h a n g e = \frac{P b o u n d (R \neq 0)}{P b o u n d (R = 0)} = \frac{1 + \frac{P}{N_{n s}} e^{- \frac{∆ ε_{p d}^{S}}{K_{b} T}}}{1 + \frac{P}{N_{n s}} e^{- \frac{∆ ε_{p d}^{S}}{K_{b} T}} + (\frac{R}{N_{n s}} e^{- \frac{∆ ε_{d}^{r}}{K_{b} T}})^{n}}$

R is the number of repressor proteins and ∆ε_d^r the specific binding energy of the repressor proteins, the same combinatorics rules apply repressor binding as for polymerase binding. n is the number of repression regulators should those regulators have the same binding energy. Because they act independently on whether or not an RNA polymerase can bind, the regulatory values are multiplied. The equation can now be incorporated into the model describing the Kill-Switch by tying production rates to the fold change of gene expression (K_prod x Fold change), i.e. the production rate will decrease proportionally to changes in gene expression.

Model Equations

Ordinary differential equations describe the genetic circuit, three of based on the statistical mechanics approach. The Deg_protein and Dil_div terms represent the half-life and dilution rate of the proteins. Because the third equation (the CIλ production) relies on an external input and is not dependent on any of the other equations, the ordinary differential equations uses Michaelis-Menten kinetics to simulate a fusaric acid input. In every simulation fusaric acid is inputted at t = 10 hours [1,2,10].

$\frac{d L a c I}{d t} = K_{p r o d_{L a c I}} \frac{1 + \frac{P}{N_{n s}} e^{- \frac{∆ ε_{p d}^{S}}{K_{b} T}}}{1 + \frac{P}{N_{n s}} e^{- \frac{∆ ε_{p d}^{S}}{K_{b} T}} + (\frac{A V T e t R}{N_{n s}} e^{- \frac{∆ ε_{d}^{T e t R_L a c}}{K_{b} T}})^{n T e t_L a c}} {- D e g}_{L a c I} L a c I - {D i l}_{d i v} L a c I$

$\frac{d C I λ}{d t} = K_{p r o d_{C I λ}} \frac{[f u s a r i c a c i d]}{[f u s a r i c a c i d] + K_{d}} - {d e g}_{C I λ} C I λ - {D i l}_{d i v} C I λ$

$\frac{d T o x i n}{d t} = K_{p r o d_{L a c I}} \frac{1 + \frac{P}{N_{n s}} e^{- \frac{∆ ε_{p d}^{S}}{K_{b} T}}}{1 + \frac{P}{N_{n s}} e^{- \frac{∆ ε_{p d}^{S}}{K_{b} T}} + (\frac{A V T e t R}{N_{n s}} e^{- \frac{∆ ε_{d}^{T e t R_{t o x i n}}}{K_{b} T}})^{n T e t_{T o x i n}} (\frac{A V C I λ}{N_{n s}} e^{- \frac{∆ ε_{d}^{C I λ t o x i n}}{K_{b} T}})^{n C I λ_G F P}} {- d e g}_{T o x i n} T o x i n - {D i l}_{d i v t o x i n}$

Software used

All models were made in Python (Python software foundation, Delaware, US) version 2.71 for Mac OS X version 10.5.8. Solving of ODE systems was done using the standard soln function in Python

Results

System Behaviour and Optimal Repressor Binding Site Configuration

Before the optimal configuration of repressor binding sites could be identified, the behaviour of the system had to be assigned scores representing a functioning or non-functioning Kill-switch. Three distinct behaviours could be identified:

0: The Kill-switch does not work, the toggle switch is out of balance and does not function.

1: The system performs less efficiently, though the toggle switch changes state, the toxin promoter is leaky.

2: The system performs to design, after a fusaric acid input the toggle switch changes state and toxin is produced when CIλ leaves the system.

In total eight (A,B,..,H) different configurations of repressor binding sites were tested (Figure 4). Two: F and H, were found to be produce Kill-switch systems robust towards variations in binding strengths and production rates. The result of the best scoring configuration H has been magnified (figure 5). The repressor binding site configuration corresponding to this map has the highest chance of working in an experimental set-up. The x-axis shows the fold change in protein production, the y-axis has increased binding strengths for individual repressors by an order of magnitude each row. The color map indicates that increases in repressor binding strengths can be compensated for by increased production rates of either the target gene or decreased production of the repressor itself. The practical implications for constructing the an optimized version of the Kill-switch were as follows: two TetR repressor binding sites on the LacI promoter, two LacI and one CIλ repressor binding site on the TetR promoter, two TetR and one CIλ repressor binding site on the toxin promoter.

**Figure 4.** Color maps indicating functioning and non-functioning systems. Each letter represents different repressor binding site configurations. Each small square within the color maps represents a score for a simulation of the system with a unique set of parameters. The colors correspond to a working system (2:red), leaky system (1:green) and a non-functioning system (0:blue). The table underneath shows the configuration of the repressor binding sites. The type of repressor binding site is specified followed by the promoter it is located on. The letters correspond to the colormaps

**Figure 5.** The best scoring repressor binding site configuration H. Each small square within the colour maps represents a score for a simulation of the system with a unique set of parameters. The x-axis shows the production rates in uM/min. The Y-axis has the repressor binding strengths. Using the lowest value found in literature [11], each row the value for th repressor binding strength increases by an order of magnitude.

Statistical Mechanics in Practice

There were two main reasons for using statistical mechanics:

(I) Statistical mechanics provides a way describe the system in terms of the number of repressors and repressor binding sites, their binding affinity and the position of the repressor binding on the promoter [1,2].

(II) In contrast to Michaelis-Menten kinetics, statistical mechanics offers more freedom studying the behavior of differently designed promoters.Michaelis-Menten requires the dissociation constants of the repressor proteins to be experimentally determined [1,2].

Based on the results of the model the experimentalists ordered oligonucleotides and assembled six promoters. Building two Kill-switches in parallel the chance of constructing a stable system increases. The assumption that the system is optimized and stable has been incorporated in the model quantifying stability .

System Cost

Introduction

BananaGuard was designed without knowing whether our engineered Pseudomonas putida would be outcompeted by other bacteria and fungi found in the rhizosphere of banana roots [12, 13]. For a successful application it is crucial that our strain can survive and compete near the banana plant roots. This seemed challenging as our system consists of multiple constitutively expressed genes controlling the kill switch and toxin-antitoxin systems. Moreover, when the system is switched to the active state by sensing fusaric acid, BananaGuard produces multiple fungal growth inhibitor enzymes and compounds. The The results display the intrinsic behavior of Kill-switch stability and performance.

integrated synthetic pathways use metabolic resources that would otherwise be dedicated to cellular maintenance or growth. The synthetic pathways therefore reduce the bacteria's potential to sustain themselves in the rhizosphere. Because of this reduced potential we wondered to what degree P. putida metabolism was affected by the introduced synthetic genes and if need be, to what extent the system could be changed to lessen the impact. To approach this problem we used a genome-scale metabolic model (GSMM) [14]. Our GSMM extended by the BananaGuard system predicts the interaction of our integrated plasmids with genes originating from endogenous pathways. With this prediction of the metabolic stress we gain insight in the metabolic burden caused by the integration of our plasmids. This will give an indication about BananaGuard if it will lose its capability to compete with other rhizosphere-populating micro-organisms in its resting state.
Continue to learn more about a genomic-scale metabolic model or view the results.

Theory and experimental design

In order to answer our questions with a GSMM we need a model that:

contains a comprehensive representation of P. putida metabolism
is capable of predicting growth rates in various environments
includes all reactions that are important for the BananaGuard system

For this reason a comparison was made between various available GSMMs for P. putida. We took a in-depth look in two models that were up-to-date, published and verified: iJP962 and iJN746[15,16]. For the necessary work to model the metabolic stress in P. putida, the best starting model needs to be chosen which has the most potential for accurately modeling the metabolic stress of our system.

System Performance

Introduction

In order to gain insight in the performance of our biological control agent in the soil the optimized system was modelled over time. Focusing on the robustness of the Kill-switch in the presence of fluctuating protein concentrations and the activation of the Kll-switch upon a fusaric acid influx, the following system characteristics were taken into account:

As a bi-stable system the kill-switch is subject to both extrinsic noise factors e.g. cell division and intrinsic noise e.g. fluctuations in repressor proteins. This affects the stability of the toggle-switch within the Kill-switch and thereby the system as a whole. Large fluctuations in the toggle-switch repressor protein concentrations will cause the Kill-switch to change state intermittently (Noise). The unwanted production of CIλ caused by leaky promoters is another factor contributing to kill-switch instability. A high basal production rate for CIλ will weigh the toggle-switch in favour of the fungal growth inhibitor producing LacI state (Leakiness). A state that is vital for our biological control agent to work but only when induced in the presence of F. oxysporum oxysporum (Induction). To that end noise, leakiness and an influx of fusaric acid are the focal points in a stochastic model of our genetic circuit. The model aims to quantify Kill-switch stability and characterize BananaGuards performance in the presence of F. oxysporum. The model takes into account cell division dividing the cellular contents the moment the cell has doubled in size. The average growth rate provided by the metabolic model and the impact of anti-gene transfer toxin antitoxin system on cell growth rate (figure 8) [34,35,37].

**Figure 8.** A schematic overview of the components contained within the model: inductuction of the fusaric acid promoter and thereby the system, stability of the kill-switch and impact of the toxin anti-toxin system on cell growth and population dynamics. .

Theory and experimental design

Stochastic Modeling and Toxin-Antitoxin Systems

The performance of BananaGuard is defined by its ability to destroy F. oxysporum upon detection, its ability to compete with other micro-organisms and the stability of the genetic circuit. The ability to destroy F. oxysporum has been tested in vitro. The biological control agents’ ability to compete with other soil born micro-organisms has been modeled in the metabolic model. The stability of the genetic switch will be explored in the following model, there are three key factors in assessing this aspect of the biological control agent: Noise, Leakiness and Induction (figure 9)

Noise (I), or variability of gene expression is of two kinds: 'intrinsic' noise derives from the natural variability in the way that molecules interact. This follows the observed difference in protein expression levels for identically regulated biological systems. 'Extrinsic' noise arises from random fluctuations in the environment or differences in the cellular sate. This comes in part from local pressure and temperature variations or the unequal distribution of cellular organelles after cell division1. Our biological control agent will have to perform in these noisy environments. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. In bi-stable systems given a sizeable noise term, prolonged exposure to a single state increases the chance of a state change. These phenomena could be detrimental to the functioning of our biological control agent. If a large part of the BananaGuard population changes from the resting state (not induced by fusaric acid) to the fungal growth production state without an external input, the system would not function. The following model therefore aims to quantify the stability of our biological control agent. If a large part of the population is robust to noisy environments we can confirm the functionality of the system [34,35].

Leakiness (II) is another factor to contributing to the instability of our bi-stable system. Most toggle switch dependent systems do not include a third interaction factor interfering with the stability of one state. The repressed CIλ on the input/output plasmid of the Kill-switch will not be completely repressed. Biologically a repressor does not stay bound, there is a constant turnover of the repressor protein on the promoters. If in the very short timeframe of promoter non-occupancy a RNA polymerase binds, CIλ will be transcribed. The binding of RNA polymerases occurs intermittently and leads to a basal production rate of CIλ. A high enough basal production rate weighs the toggle-switch in favour of the fungal inhibiting LacI state, in effect stopping the biological control agent from functioning. The model therefore aims to investigate the tolerance levels for the leakiness of CIλ and characterize the conditions in which it has the greatest impact [37].

Induction (III) ties into the performance aspect of our biological control. The biological control agent has to perform in the presence of F. oxysporum. It is imperative that the Kill-switch system is activated upon an influx of fusaric acid. In the soil the influx of fusaric acid will not be continuous but rather interlaced with periods low induction. Most likely leading to an increasing but haggard production of CIλ. The model therefore aims to characterize the optimal conditions for the successful activation of the fungal growth inhibitor producing state [34].

Toxin-Antitoxin Systems

Toxin-antitoxin systems in micro-organisms can act as a survival mechanism for micro-organisms in stressful environments. Operating as an hysteric switch, a small percentage of the population enters a toxin producing state. In this state protein production and cell growth are inhibited causing the cells to become persistor cells. This ‘metabolic inhibition’ makes these persistor cells robust to unfavorable conditions in the environment. Fluctuations in the toxins produced interfere with both the growth rate and protein translation of our biological control agent. This is therefore an important aspect of the stochastic model. Combining the Kill-switch with the toxin antitoxin system in a model over cell division will give insight in the effects of varying cell division times on the different destabilizing factors and the induction of the biological control agent by fusaric acid. It will also outline the population dynamics of our biological control agent. It is important to note that our chassis P. putida already has six native toxin-antitoxin systems. These have been incorporated into the model [38,39].

Model Equations

The model consists of eleven differential equations. The first six equations describe the dynamics of the different repressor proteins, in the form of a Hill function. All proteins have an equation describing RNA transcription (equations) and translation (term I), protein degradation (term II) and the stochastic noise term (d = 0.2) (term III). The term below the translation describes the impact of the free proteolytic proteins from the toxin anti-toxin system. Should the level of free toxin increase the value below the fraction will reduce protein translation efficacy [10,38,40,41].

$\frac{d L a c I_{R N A}}{d t} = \frac{P r o d_{L a c I_R N A}}{1 + (\frac{T e t R}{K_{T e t R}})^{n_{T e t R}}} - d e g_{L a c I_R N A} L a c I_{R N A}$

$\frac{d L a c I}{d t} = \frac{T r a n s_{L a c I_R N A} L a c I_{R N A}}{1 + \frac{T o x i n^{n}}{K T_{2}^{n}}} - d e g_{L a c I} L a c I + ζ_{L a c I}$

$\frac{d {T e t R}_{R N A}}{d t} = \frac{P r o d_{T e t R}}{1 + (\frac{L a c I}{K_{L a c I}})^{n_{T e t R}} + (\frac{λ C I}{K_{C I λ}})^{n_{C I λ}}} - d e g_{T e t R_R N A} T e {t R}_{R N A}$

$\frac{d T e t R}{d t} = \frac{T r a n s_{T e t R_R N A} T e t R_{R N A}}{1 + \frac{T o x i n^{n}}{K T_{2}^{n}}} - d e g_{T e t R} T e t R + ζ_{T e t R}$

$\frac{d C {I λ}_{R N A}}{d t} = \frac{F u s a r i c A c i d P r o d_{C I λ}}{F u s a r i c A c i d K_{F u s a r i c A c i d}} - d e g_{C I λ_R N A} C I λ_R N A$

$\frac{d C I λ}{d t} = \frac{T r a n s_{C I λ_R N A} C {I λ}_{R N A}}{1 + \frac{T o x i n^{n}}{K T_{2}^{n}}} - d e g_{C I λ_R N A} C I λ_R N A$

The toxin anti-toxin systems can be described by two equations taken from the paper ‘Molecular mechanisms of multiple toxin–antitoxin systems are coordinated to govern the persister phenotype’ by Rick A. Fasani and Michael A. Savageau (2013). They proposed a mathematical description of a hysteric switch, linking it to fluctuating levels in free toxins and antitoxins and the growth rate. If the levels of the free toxin increases the growth rate of the cell decreases. Protein dilution caused by the growth rate was the main factor of protein degradation. This means that if a balanced is tipped the cell changes state and becomes a slow growing persistor cell.

The three equations that follow describe the production of toxin RNA as part of the Kill-switch and free toxins and antitoxins(term I), active degradation (term II), and stochastic fluctuations (term III). The equation describing free toxins has a RNA translation term added. The production terms for the free toxins and antitoxins have two components in the denominator. The first term describes the inhibition of protein synthesis caused by the free toxins. The second term describes auto repression of the toxin antitoxin genes by the toxin anti-toxin complexes that are formed [38].

$\frac{d T o x i n_{R N A}}{d t} = \frac{P r o d_{T o x i n}}{1 + (\frac{C I λ}{K_{C I λ}})^{n_{C I λ}} + (\frac{T e t R}{K_{T e t R}})^{n_{T e t R}}} - d e g_{T o x_{R N A}} T o x i n_{R N A}$

$\frac{d T o x i n}{d t} = \frac{α}{(1 + \frac{T o x i n^{n}}{K T_{2}^{n}}) (1 + \frac{A n t i T o x i n^{s}}{K p_{1}^{s}} + (2 \frac{A n t i t o x i n^{s} T o x}{K p_{2}^{s}})^{P} + (\frac{A n t i t o x i n^{s} T o x^{s}}{K p_{1}^{s} K H^{s}}))} + \frac{T r a n s_{T o x i n_R N A} T o x i n}{1 + \frac{T o x i n^{n}}{K T_{2}^{n}}} - d e g_{T o x i n} + ζ_{T o x i n}$

$\frac{d A n t i T o x i n}{d t} = \frac{α σ}{(1 + \frac{T o x i n^{n}}{K T_{2}^{n}}) (1 + \frac{A n t i T o x i n^{s}}{K p_{1}^{s}} + (2 \frac{A n t i t o x i n^{s} T o x}{K p_{2}^{s}})^{P} + (\frac{A n t i t o x i n^{s} T o x^{s}}{K p_{1}^{s} K H^{s}}))} - d e g_{A n t i T o x i n} A n t i T o x i n + ζ_{A n t i T o x i n}$

dV/dt represents cell growth. Cells grow exponentially at a rate of µ_max, the stochastic fluctuation of free toxin levels inhibit maximum growth. The model states that given an initial volume of 1, cell volume will increase according to µV where µ is µ_max divided by the free toxin inhibition term [38]. The moment the cell has doubled in size (volume = 2), the cell contents are divided over the daughter cells following a binomial distribution. The simulation is restarted following one of these daughter cells. Should the kill-switch change state the initial conditions are reset to those at the start of the simulation [42].

$\frac{d V}{d t} = \frac{µ_{m a x} V}{1 + \frac{T o x i n^{n}}{K T_{1}^{n}}}$

The last equation describes the concentration of fusaric acid in the cell. Not much is known about fusaric acid dynamics besides measured quantities in the soil after infection. The fusaric acid detector experiments have given an indication with respect to the kinetics of the fusaric acid promoter. The required number of fusaric acid molecules for promoter activation is quite low, especially compared to the levels of fusaric acid that have been measured in the soil after infection. The influx is stochastic and is weighed such that the concentration increases over time. This randomized increasing influx represents the varying influx of fusaric acid into our BananaGuard population.

$\frac{d F u s a r i c A c i d}{d t} = ζ_{I n f l u x} - d e g_{F u s a r i c A c i d} F u s a r i c A c i d$

Model Sensitivity

Below a table for the parameter tolerances for the maintenance bi-stability of the kill-switch and functionality in the toxin antitoxin system. A local sensitivity analysis was done in order to find the ranges of the bi-stable region for this parameter set by running simulation in a set state of the Kill-Switch. If the parameters cause instability, the state at the end if the simulation changed. The values for the toxin-antitoxin tolerances were published by Rick A. Fasani and Michael A. Savageau (2013). The indicate the parameter ranges where hysteric switching occurs

Table 2. The parameter tolerances for the maintenance of a bi-stable Kill-switch and hysteric switching in the toxin-antitoxin system .

Software used

All models were made in Python (Python software foundation, Delaware, US) version 2.71 for Mac OS X version 10.5.8. Solving of ODE systems was done using the standard soln function in Python

Results

The results follow the trend of Noise, Leakiness and input. The first set of figures describe the Kill-switch system over cell division without the presence of the toxin anti-toxin system, this means the cell division times are equal for all simulations. The results display the intrinsic behavior of Kill-switch stability and performance.

The second set of figures follows the model with the toxin-antitoxin system included. The varying growth and protein production rates caused by fluctuations in the free toxin concentration do impact the functionality but the overall system performs in line with the results obtained (Figure 10) [38].

Noise, A: The significant increase in stability given a longer cell division time coincides with the fact that rapid protein dilution adds to the instability of the Kill-switches. Biologically the ratio between the protein levels at steady state has decreased, the chance that the noise term will have an impact therefore increases, resulting in a higher number of Kill-switches that have changed state.

Leakiness, B: The significant increase in instability given a longer cell division and a leaky promoter can be explained in a reversed but similar manner to figure 1A. Lower dilution rates due to increased division times increases the amount of CIλ in the cell. Biologically, a more effective repression of the TetR promoter occurs due higher levels of CIλ. This lowers the steady state levels of the TetR protein decreasing the ratio of the protein levels at steady state, increasing the chance that the noise term will have an impact.

Induction, C: The plots correspond production rate and the activation of the CIλ repressor protein to be limiting to the performance of the kill-switch. Similar to the effect of the basal production of CIλ, the increased division times and thereby lowered dilution rates allow CIλ to accumulate. The accumulation gradually lowers the steady state ratios of the toggle switch proteins until the system changes state. The results indicate that for a parameter set aiming to describe low copy plasmid numbers, the performance of our biological control agent genetic circuitry is tied to its growth rate.

Overall the results for the Kill-switch modeled over cell division have shown that a fast growth rate affects the ‘basal’ stability of the genetic circuit. High dilution rates increase the chance of instability due to noise. Leakiness and induction in and of the system are more likely to be effective at a slow growth rates. In both cases the build-up of CIλ over time is key in accessing the systems instability and performance.

The stability and performance of the Biological Control Agent over Cell Division

The following results will outline the following:

(I) The impact of the toxin-antitoxin anti-gene gene transfer system on cell growth.

(II) The impact of the toxin antitoxin on promoter leakiness and induction.

For both Leakiness and induction, the maximum and the predicted growth rate was simulated. The prior in order to ascertain the general behavior of the system, the latter in order to get a more accurate picture of our biological control agents performance in the soil (figure 11). The effect of just noise on the stability of the biological control agent has been neglected because its effect was statistically insignificant.

Figure 11. The figures show the stability given a leaky promoter . Taking the form of a histogram, the y-axis shows the number of cell divisions that have occurred at a specific time point. The X-axis shows the number of minutes it takes for the cell to divide. The light green bars indicate a functioning system and a specific cell division time, coloured bars indicate non-functioning systems corresponding to a specific basal production rate.

(A) For a maximum growth rate corresponding to a division time of 45 minutes, a basal CIλ production of 50 nM/min or higher destabilizes the kill-switch in slow growing cells. The frequency of division times within the population of the cells is spread out. A spread over longer divsion times can be observed after an initial peak at 60 minutes. Protein dilution due to cell division is high, the population dynamics is not impacted. A total of 20000 simulation were run.

(B) For a maximum growth rate corresponding to a division time of 180 minutes a basal CIλ production of 50 nM/min or higher destabilizes the Kill-switch. The population dynamics are affected. Low protein dilution due to slow growth causes the Kill-switch to leak toxin. Higher basal production rates compensate, increasing the average growth rate but also the instability. A total of 5000 simulation were run.

Leakiness, A: The population dynamics are effected by free toxins from the toxin-antitoxin system. For a lower maximum growth rate and therefore a lower protein dilution rate the fluctuations in the level of free toxins is attributed to more than just stochasticity. Because a toxin is included on the kill-switch its promoter is subject to leakiness with a low amount of repressor protein present. With lower dilution rates, low amounts of CIλ are able to build up slower over time. The CIλ protein represses both the toxin and the TetR promoter on the toggle switch. This results in lower levels of TetR for which the CIλ cannot compensate over the course of its build-up. A deficiency that translates in a leaky toxin promoter causing a portion of the cells to distribute over longer division times. The effect of leakiness on the stability of the genetic circuit corresponds to the results obtained in figure 1.B. Over the total number simulations only 13% of the kill-switches became unstable given a basal production rate larger than 50nM and a maximum growth rate of 45 minutes. As expected all instances of instability occurred in the slow growing portion of the cell population.

Leakiness, B: Because the population of cells averaged a significantly higher growth rate the percentage of unstable kill-switches increased proportionally too, once again confirming increased instability for lengthy cell divisions cycles. With respect population dynamics, the same logic that is applied to figure 1B applies to figure 1A. the effect however of a leaky CIλ can clearly be seen. A combination of an included toxin on the kill-switch, a promoter subject to leakiness and Lower dilution rates increase the levels of free toxin. Low basal production levels for CIλ increases TetR repression, alleviating toxin repression for which the CIλ cannot compensate if its concentration is to low. Higher levels of basal CIλ production reduce the levels of free toxins speeding up cell growth but decreasing stability. In order to avoid this problem the basal production rate of CIλ needs to be measured and adjusted if necessary, values lower than 50nM for the basal production appear to have no significant effect on the stability of the genetic circuit. For a low basal production of CIλ entails a low basal production of toxin.

Figure 12. AThe figures show the performance of our biological control agent upon induction. All plots take the form of histograms, the y-axis shows the number of cell divisions that have occurred at a specific time point over 20000 simulations. The X-axis shows the number of minutes it takes for the cell to divide. The green bars indicate a functioning system and a specific cell division time, red bars indicate non-functioning systems given a specific cell division time. Two maximum growth rates uMax were investigated 45 minutes and 180 minutes.

(A) For a maximum growth rate of 45 minutes 36% of the kill-switches activate, longer division times activate the cells more effectively.

(B) For a maximum growth rate of 180 minutes 98% of the kill-switches activate, longer division times activate the cells more effectively.

Induction, A and B: The population dynamics for an induced system varies slightly. The same logic applies with respect to leaky toxins but the time-spans are different. Because the build-up of CIλ occurs faster in an induced system compared to a system with a basal transcription, the time frame in which the toxin promoter leaks is smaller. This results in faster average growth rate compared to a situation with leaky promoters. Induction in and of itself works in accordance with the results obtained in figure 1. Slower growth rates activate the system more efficiently if a maximum growth rate 36% of the kill-switches is assumed, given the growth rate obtained from the metabolic model an efficiency of 98% is achieved.

Conclusion

The performance of BananaGuard as a biological control agent is defined by its ability to destroy F. oxysporum upon detection, its ability to compete with other micro-organisms and the stability of the genetic circuit. The ability to destroy F. oxysporum has been tested in vitro. The modeling has therefore been used to predict the answer to the remaining two questions.

Can the BananaGuard system survive in the presence of competing micro-organisms?

Is the genetic circuit robust to biological fluctuations?

We have demonstrated that different modeling approaches can be taken and integrated with both the lab and one another. The statistical mechanics model, modeling promoter design has shown that a prediction regarding the functionality of the kill-switch can be made with respect to the different constituents on the promoter. The predicted configuration of repressor binding sites has been used in designing a second, optimized version of the kill-switch. The model estimating the cost of the introduced genetic circuit has shown that in its resting state, BananaGuard is capable of competing with other micro-organisms in the soil on a metabolic level. The average growth rate obtained in this model has been used in the final model describing the stability and performance of BananaGuard. The results obtained have characterized several aspects of the kill-switch and the toxin anti-toxin system. First, noise can impact the stability of the kill-switch to a certain degree. Secondly, different basal production levels of CIλ can have different effects on population dynamics, cell growth and the stability of the kill-switch, a point of attention for final construct of the system. Finally, the kill-switch will perform with 98% efficiency given the slow growth rate in the soil predicted by the metabolic model.

Continue to Interlab Study >>

Appendix

Table of parameters

Promoter Design

Designation Value Description Reference

K_{prod_LacI} 1 µM min^-1 LacI production rate [9]

K_{prod_TetR} 1 µM min^-1 TetR production rate [9]

K_{prod_CIλ} 1 µM min^-1 CIλ production rate [9]

K_{prod_RNAtoxin} 0.75 µM min^-1 Toxin production rate [9]

Dil 0.023 min^-1 Degradation due to cell division [9]

deg_LacI 0.034 min^-1 Degradation (Half-life LacI) [3,9]

deg_TetR 0.023 min^-1 Degradation (Half-life TetR) [4,9,43]

deg_CIλ 0.0011 min^-1 Degradation (Half-life CIλ) [4,9,43]

deg_Toxin 0.0025 min^-1 Degradation (Half-life Toxin) [5]

A 6.02e²³ Avogadro’s number [9]

V 1.5e^-16 dm^-3 Cell volume [9]

P 1900 Number of RNA polymerases [9]

N_ns 500,000 Non-specific binding sites [1,2,9]

K_b 1.38e^-23m²kg s^- Boltzmann constant [9]

T 298 Temperature in Kelvin -

∆ε_pd^S 0.01 nM Energy of the state, RNA polymerases [1,2]

∆ε_d^TetR_LacI 100 pM Energy of the state, TetR repressor [4,6,9]

∆ε_d^TetR_Toxin 100 pM Energy of the state, TetR repressor [4,6,9]

∆ε_d^LacI_TetR 1 pM Energy of the state, LacI repressor [3,9,43]

∆ε_d^CIλ_TetR 1 pM Energy of the state, CIλ repressor [4,9]

∆ε_d^CIλ_Toxin 1 pM Energy of the state, CIλ repressor [4,9]

nTet_LacI - Repressor binding sites TetR binding PLacI -

nTet_Toxin - Repressor binding sites TetR binding PToxin -

nCIλ_Toxin - Repressor binding sites CIλ binding PToxin -

nCIλ_TetR - Repressor binding sites CIλ binding PTetR -

nLacI - Repressor binding sites Laci binding PTetR -

System Cost

Designation Value Description Reference

model iJP962 - Genomic-scale metabolic model [6,18]

carbonsource - Fructose, Glucose, Alanine, Asparagine, Aspartate, L-Leucine, Serine, L-Threonine, L-Proline, D-Glutamate [15]

glucuptake 4.43 mmol gDW^-1hr^-1 Carbon uptake rate [1]

NGAM 3.96 mmol gDW^-1hr^-1 Non-growth maintenance factor [6]

KS_rate 0.2 μM min^-1 Protein production rate Promoter Design

TAT_rate 1 nM min^-1 Toxin/anti-toxin production rate Promoter Design

Cell density 0.040 gDW l^-1 gDW/OD600=0.1 [20]

ncells 1e⁸ cells ml^-1 cells/OD600=0.1 [21]

Chitinase 0.2 μM min^-1 Chitinase production rate [7] and Promoter Design

Pyoverdine 150 μmol gDW^-1hr^-1 Pyoverdine production rate Experiment

DAPG 25 μmol gDW^-1hr^-1 DAPG production rate [8]

DMDS 0.845 μmol gDW^-1hr^-1 DMDS production rate [9]

Nplasmids 10 plasmids cell^-1 Number of plasmids per cell (low copy number) [23]

plasmidTD 0.3 h^-1 Time needed for at least a doubling of the number of plasmids per cell [1]

TDrhizo 3 hr Doubling time P. putida in the rhizosphere [1]

System Performance

Designation Value Description Reference

K_{prod RNA_LacI} 0.75 µM min^-1 LacI production rate [9,11]

K_{prod RNA_TetR} 0.75 µM min^-1 TetR production rate [9,11]

K_{prod RNA_CIλ} 0.75 µM min^-1 CIλ production rate [9,11]

K_{prod_RNAtoxin} 0.75 µM min^-1 Toxin production rate [9,11]

deg_ToxinRNA 0.178 min^-1 Degradation (Half-life Toxin RNA) [5]

deg_RNA 0.82 min^-1 Degradation (Half-life RNA) [45]

deg_LacI 0.034 min^-1 Degradation (Half-life LacI) [3,44]

deg_TetR 0.023 min^-1 Degradation (Half-life TetR) [43]

deg_CIλ 0.011 min^-1 Degradation (Half-life CIλ)

deg_Toxin 0.000155 min^-1 Degradation (Half-life Toxin) [38]

deg_Anti-Toxin 0.0011 min^-1 Degradation (Half-life Anti-Toxin) [38]

Trans_LacI 0.12 µM min^-1 Translation Rate LacI [9,45]

Trans_TetR 0.08 µM min^-1 Translation Rate TetR [9,45]

Trans_CIλ 0.12 µM min^-1 Translation Rate CIλ [9,45]

Trans_Zeta 0.12 µM min^-1 Translation Rate Zeta [9,45]

K_TetR 4 Dissociation constant (TetR) [4,6,9,43]

K_LacI 1.75 Dissociation constant (LacI) [3,9,11]

K_CIλ 4 Dissociation constant (CIλ) [4,43]

N_TetR 3 Hill constant(TetR) [1,2,9,10]

N_LacI 2 Hill constant(LacI) [3,9,43]

K_CIλ 2 Hill constant(CIλ) [4]

α 1 nM min^-1 Free Toxin-Antitoxin production [38]

σ 10 Antitoxin factor [38]

μ_max 0.0153-0.0038 Maximum cell growth rate Metabolic Model

KP₁ 1000 nM Dissociation constant(Toxin Repression) [38]

KP₂ 10 nM Dissociation constant(antiToxin Repression) [38]

KT₁ 10 Dissociation constant(Growth inhibition) [38]

KT₂ 100 Dissociation constant(Translation inhibition) [38]

n 6 Half of the total number of Toxin-Antitoxin systems [38]

p 2 scaling factor [38]

KH 1000 nM Dissociation constant(Antitoxin) [38]

References

Promoter Design

Bintu L, Buchler NE, Garcia HG, Gerland U, Hwa T, Kondev J,Kuhlman T, Phillips R (2005): Transcriptional regulation by the numbers: models. Curr Opin Genet Dev, 15:125-135.

Bintu L, Buchler NE, Garcia HG, Gerland U, Hwa T, Kondev J,Kuhlman T, Phillips R (2005): Transcriptional regulation by the numbers: application. Curr Opin Genet Dev (2), 15:125-135.

Mitchel Lewis (2005) The Lac repressor. C. R. Biologies 328 (2005) 521–548.

Subhayu Basu1, Yoram Gerchman, Cynthia H. Collins3, Frances H. Arnold& Ron Weiss,2: A synthetic multicellular system for programmed pattern formation. Nature 434 ,28 April 2005, 1130-113.

Falcon C.M and Matthews K.S. (2000) Operator DNA sequence Variation Enhances High Affinity Binding by Hinge Helix Mutants of Lactose Repressor Protein. Biochemistry. 39, 11074-11084.

Christian Berens , Dirk Schnappinger and Wolfgang Hillen ; The Role of the Variable Region in Tet Repressor for Inducibility by Tetracycline, Biological Chemisttry Volume 272, Number 11, Issue of March 14, 1997 pp. 6936-6942.

ME346A Introduction to Statistical Mechanics – Wei Cai – Stanford University – Win 2011, Homepage; micro.stanford.edu/.

Lee T.I, Young R.A: Transcription of eukaryotic protein-coding genes. Annu Rev Genet 2000, 34:77-137. p.109.

Milo et al. Nucl. Acids Res. (2010) 38: D750-D753.

Timothy S. Gardner, Charles R. Cantor1 & James J. Collins; Construction of a genetic toggle switch inEscherichia coli, 20 January 2000, Nature 403, 339-342.

iGEM, Aberdeen Schotland 2009, homepage: igem.org.

System Cost

Bennett, R. A., & Lynch, J. M. (1981). Bacterial growth and development in the rhizosphere of gnotobiotic cereal plants. Journal of General Microbiology, 125(1), 95-102.

Molina, L., Ramos, C., Duque, E., Ronchel, M. C., Garcı́a, J. M., Wyke, L., & Ramos, J. L. (2000). Survival of< i> Pseudomonas putida KT2440 in soil and in the rhizosphere of plants under greenhouse and environmental conditions. Soil Biology and Biochemistry, 32(3), 315-321.

Puchałka J, Oberhardt MA, Godinho M, Bielecka A, Regenhardt D, et al. (2008) Genome-Scale Reconstruction and Analysis of the a KT2440 Metabolic Network Facilitates Applications in Biotechnology. PLoS Comput Biol 4(10): e1000210. doi: 10.1371/journal.pcbi.1000210.

Oberhardt, M. A., Puchałka, J., Dos Santos, V. A. M., & Papin, J. A. (2011). Reconciliation of genome-scale metabolic reconstructions for comparative systems analysis. PLoS computational biology, 7(3), e1001116.

Nogales, J., Palsson, B. Ø., & Thiele, I. (2008). A genome-scale metabolic reconstruction of Pseudomonas putida KT2440: iJN746 as a cell factory. BMC systems biology, 2(1), 79.

van Duuren, J. B., Pucha, J., Mars, A. E., Bücker, R., Eggink, G., Wittmann, C., & dos Santos, V. A. (2013). Reconciling in vivo and in silico key biological parameters of Pseudomonas putida KT2440 during growth on glucose under carbon-limited condition. BMC biotechnology, 13(1), 93.

The UniProt Consortium,Activities at the Universal Protein Resource (UniProt),[Pyocin,Q88ID3].

Schnider-Keel, U., Seematter, A., Maurhofer, M., Blumer, C., Duffy, B., Gigot-Bonnefoy, C., ... & Keel, C. (2000). Autoinduction of 2, 4-diacetylphloroglucinol biosynthesis in the biocontrol agent Pseudomonas fluorescensCHA0 and repression by the bacterial metabolites salicylate and pyoluteorin. Journal of Bacteriology, 182(5), 1215-1225.

Arfi, K., Landaud, S., & Bonnarme, P. (2006). Evidence for distinct L-methionine catabolic pathways in the yeast Geotrichum candidum and the bacterium Brevibacterium linens. Applied and environmental microbiology, 72(3), 2155-2162.

Salah El Din, A. L. M., Kyslík, P., Stephan, D., & Abdallah, M. A. (1997). Bacterial iron transport: Structure elucidation by FAB-MS and by 2D NMR (H¹,C¹³,N¹⁵) of pyoverdin G4R, a peptidic siderophore produced by a nitrogen-fixing strain of Pseudomonas putida. Tetrahedron, 53(37), 12539-12552.

Boukhalfa, H., Reilly, S. D., Michalczyk, R., Iyer, S., & Neu, M. P. (2006). Iron (III) coordination properties of a pyoverdin siderophore produced by Pseudomonas putida ATCC 33015. Inorganic chemistry, 45(14), 5607-5616.

Matlab

Cobra Toolbox

Gurobi

Buxton, E. W. (1962). Root exudates from banana and their relationship to strains of the Fusarium causing Panama wilt. Annals of Applied Biology, 50(2), 269-282.

Bais, H. P., Weir, T. L., Perry, L. G., Gilroy, S., & Vivanco, J. M. (2006). The role of root exudates in rhizosphere interactions with plants and other organisms. Annu. Rev. Plant Biol., 57, 233-266.

Armstrong, W., & Drew, M. C. (2002). Root growth and metabolism under oxygen deficiency. Plant roots: the hidden half, 3, 729-761.

Oberhardt, M.A., J. Puchalka, V.A.P. Martins dos Santos, and J.A. Papin. 2011. Reconciliation of genome-scale metabolic reconstructions for comparative systems analysis. PLoS Computational Biology, 7(3).

BNID 107924, Milo et al 2010.

BNID 100985, Milo et al 2010.

Gross, H., & Loper, J. E. (2009). Genomics of secondary metabolite production by Pseudomonas spp. Natural product reports, 26(11), 1408-1446.

Stoker, N. G., Fairweathe, N. F., & Spratt, B. G. (1982). Versatile low-copy-number plasmid vectors for cloning in Escherichia coli. Gene, 18(3), 335-341.

System Design

Shu C-C, Chatterjee A, Dunny G, Hu W-S, Ramkrishna D (2011) Bistability versus Bimodal Distributions in Gene Regulatory Processes from Population Balance. PLoS Comput Biol 7(8): e1002140.

Volfson, D. & Marciniak, J. et al.(2005) Origins of extrinsic variability in eukaryotic gene expression. Nature 21 December 2005

Martinez Arias, A. & Hayward (2006), P. Filtering transcriptional noise during development: concepts and mechanisms. Nature Rev. Genet. 7, 34–44

Pallavi Penumetcha, Kin Lau, Xiao Zhu, Kelly Davis, Todd T. Eckdahl, A. Malcolm Campbell. (2010) Improving the Lac System for Synthetic Biology. BIOS. 7-15

R.A. Fasani, M.A. Savageau Molecular mechanisms of multiple toxin-antitoxin systems are coordinated to govern the persister phenotype (2013) . Proc. Natl. Acad. Sci. USA, 110, pp. E2528–E2537

Etienne Maisonneuve, Kenn Gerdes (2014), Molecular Mechanisms Underlying Bacterial Persisters. Cell Volume 157, Issue 3, Pages 539–548

Rué, P., & Garcia-Ojalvo, J. (2013). Modeling gene expression in time and space. Annual review of biophysics, 42, 605-627.

Swain, P. S. (2004). Efficient attenuation of stochasticity in gene expression through post-transcriptional control. Journal of molecular biology, 344(4), 965-976

Gonze, D. (2013). Modeling the effect of cell division on genetic oscillators. Journal of theoretical biology, 325, 22-33.

Braun, D., Basu, S., & Weiss, R. (2005, March). Parameter estimation for two synthetic gene networks: a case study. In Acoustics, Speech, and Signal Processing, 2005. Proceedings.(ICASSP'05). IEEE International Conference on (Vol. 5, pp. v-769). IEEE.

Purcell, O., Grierson, C. S., Bernardo, M., & Savery, N. J. (2012). Temperature dependence of ssrA-tag mediated protein degradation. J. Biol. Eng, 6(10).

Alon, U. (2006). An introduction to systems biology: design principles of biological circuits. CRC press. Nevozhay, D., Adams, R. M., Murphy, K. F., Josić, K., & Balázsi, G. (2009). Negative autoregulation linearizes the dose–response and suppresses the heterogeneity of gene expression. Proceedings of the National Academy of Sciences, 106(13), 5123-5128.

Braun, D., Basu, S., & Weiss, R. (2005, March). Parameter estimation for two synthetic gene networks: a case study. In Acoustics, Speech, and Signal Processing, 2005. Proceedings.(ICASSP'05). IEEE International Conference on (Vol. 5, pp. v-769). IEEE.

iGEM Wageningen UR 2014. Send us an email, or follow us on Twitter and Facebook!
For more information about iGEM, check out their website and Twitter!

Retrieved from "http://2014.igem.org/Team:Wageningen_UR/project/model"

[1] Bintu L, Buchler NE, Garcia HG, Gerland U, Hwa T, Kondev J,Kuhlman T, Phillips R (2005): Transcriptional regulation by the numbers: models. Curr Opin Genet Dev, 15:125-135.

[2] Bintu L, Buchler NE, Garcia HG, Gerland U, Hwa T, Kondev J,Kuhlman T, Phillips R (2005): Transcriptional regulation by the numbers: application. Curr Opin Genet Dev (2), 15:125-135.

[3] Mitchel Lewis (2005) The Lac repressor. C. R. Biologies 328 (2005) 521–548.

[4] Subhayu Basu1, Yoram Gerchman, Cynthia H. Collins3, Frances H. Arnold& Ron Weiss,2: A synthetic multicellular system for programmed pattern formation. Nature 434 ,28 April 2005, 1130-113.

[5] Falcon C.M and Matthews K.S. (2000) Operator DNA sequence Variation Enhances High Affinity Binding by Hinge Helix Mutants of Lactose Repressor Protein. Biochemistry. 39, 11074-11084.

[6] Christian Berens , Dirk Schnappinger and Wolfgang Hillen ; The Role of the Variable Region in Tet Repressor for Inducibility by Tetracycline, Biological Chemisttry Volume 272, Number 11, Issue of March 14, 1997 pp. 6936-6942.

[7] ME346A Introduction to Statistical Mechanics – Wei Cai – Stanford University – Win 2011, Homepage; micro.stanford.edu/.

[8] Lee T.I, Young R.A: Transcription of eukaryotic protein-coding genes. Annu Rev Genet 2000, 34:77-137. p.109.

[9] Milo et al. Nucl. Acids Res. (2010) 38: D750-D753.

[10] Timothy S. Gardner, Charles R. Cantor1 & James J. Collins; Construction of a genetic toggle switch inEscherichia coli, 20 January 2000, Nature 403, 339-342.

[11] GEM, Aberdeen Schotland 2009, homepage: igem.org.

[12] Bennett, R. A., & Lynch, J. M. (1981). Bacterial growth and development in the rhizosphere of gnotobiotic cereal plants. Journal of General Microbiology, 125(1), 95-102.

[13] Molina, L., Ramos, C., Duque, E., Ronchel, M. C., Garcı́a, J. M., Wyke, L., & Ramos, J. L. (2000). Survival of< i> Pseudomonas putida KT2440 in soil and in the rhizosphere of plants under greenhouse and environmental conditions. Soil Biology and Biochemistry, 32(3), 315-321.

[14] Puchałka J, Oberhardt MA, Godinho M, Bielecka A, Regenhardt D, et al. (2008) Genome-Scale Reconstruction and Analysis of the a KT2440 Metabolic Network Facilitates Applications in Biotechnology. PLoS Comput Biol 4(10): e1000210. doi: 10.1371/journal.pcbi.1000210.

[15] Oberhardt, M. A., Puchałka, J., Dos Santos, V. A. M., & Papin, J. A. (2011). Reconciliation of genome-scale metabolic reconstructions for comparative systems analysis. PLoS computational biology, 7(3), e1001116.

[16] Nogales, J., Palsson, B. Ø., & Thiele, I. (2008). A genome-scale metabolic reconstruction of Pseudomonas putida KT2440: iJN746 as a cell factory. BMC systems biology, 2(1), 79.

[17] van Duuren, J. B., Pucha, J., Mars, A. E., Bücker, R., Eggink, G., Wittmann, C., & dos Santos, V. A. (2013). Reconciling in vivo and in silico key biological parameters of Pseudomonas putida KT2440 during growth on glucose under carbon-limited condition. BMC biotechnology, 13(1), 93.

[18] The UniProt Consortium,Activities at the Universal Protein Resource (UniProt),[Pyocin,Q88ID3].

[19] Schnider-Keel, U., Seematter, A., Maurhofer, M., Blumer, C., Duffy, B., Gigot-Bonnefoy, C., ... & Keel, C. (2000). Autoinduction of 2, 4-diacetylphloroglucinol biosynthesis in the biocontrol agent Pseudomonas fluorescensCHA0 and repression by the bacterial metabolites salicylate and pyoluteorin. Journal of Bacteriology, 182(5), 1215-1225.

[20] Arfi, K., Landaud, S., & Bonnarme, P. (2006). Evidence for distinct L-methionine catabolic pathways in the yeast Geotrichum candidum and the bacterium Brevibacterium linens. Applied and environmental microbiology, 72(3), 2155-2162.

[21] Salah El Din, A. L. M., Kyslík, P., Stephan, D., & Abdallah, M. A. (1997). Bacterial iron transport: Structure elucidation by FAB-MS and by 2D NMR (H¹,C¹³,N¹⁵) of pyoverdin G4R, a peptidic siderophore produced by a nitrogen-fixing strain of Pseudomonas putida. Tetrahedron, 53(37), 12539-12552.

[22] Boukhalfa, H., Reilly, S. D., Michalczyk, R., Iyer, S., & Neu, M. P. (2006). Iron (III) coordination properties of a pyoverdin siderophore produced by Pseudomonas putida ATCC 33015. Inorganic chemistry, 45(14), 5607-5616.

[23] Matlab

[24] Cobra Toolbox

[25] Gurobi

[26] Buxton, E. W. (1962). Root exudates from banana and their relationship to strains of the Fusarium causing Panama wilt. Annals of Applied Biology, 50(2), 269-282.

[27] Bais, H. P., Weir, T. L., Perry, L. G., Gilroy, S., & Vivanco, J. M. (2006). The role of root exudates in rhizosphere interactions with plants and other organisms. Annu. Rev. Plant Biol., 57, 233-266.

[28] Armstrong, W., & Drew, M. C. (2002). Root growth and metabolism under oxygen deficiency. Plant roots: the hidden half, 3, 729-761.

[29] Oberhardt, M.A., J. Puchalka, V.A.P. Martins dos Santos, and J.A. Papin. 2011. Reconciliation of genome-scale metabolic reconstructions for comparative systems analysis. PLoS Computational Biology, 7(3).

[30] BNID 107924, Milo et al 2010.

[31] BNID 100985, Milo et al 2010.

[32] Gross, H., & Loper, J. E. (2009). Genomics of secondary metabolite production by Pseudomonas spp. Natural product reports, 26(11), 1408-1446.

[33] Stoker, N. G., Fairweathe, N. F., & Spratt, B. G. (1982). Versatile low-copy-number plasmid vectors for cloning in Escherichia coli. Gene, 18(3), 335-341.

[34] Shu C-C, Chatterjee A, Dunny G, Hu W-S, Ramkrishna D (2011) Bistability versus Bimodal Distributions in Gene Regulatory Processes from Population Balance. PLoS Comput Biol 7(8): e1002140.

[35] Volfson, D. & Marciniak, J. et al.(2005) Origins of extrinsic variability in eukaryotic gene expression. Nature 21 December 2005

[36] Martinez Arias, A. & Hayward (2006), P. Filtering transcriptional noise during development: concepts and mechanisms. Nature Rev. Genet. 7, 34–44

[37] Pallavi Penumetcha, Kin Lau, Xiao Zhu, Kelly Davis, Todd T. Eckdahl, A. Malcolm Campbell. (2010) Improving the Lac System for Synthetic Biology. BIOS. 7-15

[38] R.A. Fasani, M.A. Savageau Molecular mechanisms of multiple toxin-antitoxin systems are coordinated to govern the persister phenotype (2013) . Proc. Natl. Acad. Sci. USA, 110, pp. E2528–E2537

[39] Etienne Maisonneuve, Kenn Gerdes (2014), Molecular Mechanisms Underlying Bacterial Persisters. Cell Volume 157, Issue 3, Pages 539–548

[40] Rué, P., & Garcia-Ojalvo, J. (2013). Modeling gene expression in time and space. Annual review of biophysics, 42, 605-627.

[41] Swain, P. S. (2004). Efficient attenuation of stochasticity in gene expression through post-transcriptional control. Journal of molecular biology, 344(4), 965-976

[42] Gonze, D. (2013). Modeling the effect of cell division on genetic oscillators. Journal of theoretical biology, 325, 22-33.

[43] Braun, D., Basu, S., & Weiss, R. (2005, March). Parameter estimation for two synthetic gene networks: a case study. In Acoustics, Speech, and Signal Processing, 2005. Proceedings.(ICASSP'05). IEEE International Conference on (Vol. 5, pp. v-769). IEEE.

[44] Purcell, O., Grierson, C. S., Bernardo, M., & Savery, N. J. (2012). Temperature dependence of ssrA-tag mediated protein degradation. J. Biol. Eng, 6(10).

[45] Alon, U. (2006). An introduction to systems biology: design principles of biological circuits. CRC press. Nevozhay, D., Adams, R. M., Murphy, K. F., Josić, K., & Balázsi, G. (2009). Negative autoregulation linearizes the dose–response and suppresses the heterogeneity of gene expression. Proceedings of the National Academy of Sciences, 106(13), 5123-5128.

[46] Braun, D., Basu, S., & Weiss, R. (2005, March). Parameter estimation for two synthetic gene networks: a case study. In Acoustics, Speech, and Signal Processing, 2005. Proceedings.(ICASSP'05). IEEE International Conference on (Vol. 5, pp. v-769). IEEE.

Designation	Value	Description	Reference
K_{prod_LacI}	1 µM min^-1	LacI production rate	[9]
K_{prod_TetR}	1 µM min^-1	TetR production rate	[9]
K_{prod_CIλ}	1 µM min^-1	CIλ production rate	[9]
K_{prod_RNAtoxin}	0.75 µM min^-1	Toxin production rate	[9]
Dil	0.023 min^-1	Degradation due to cell division	[9]
deg_LacI	0.034 min^-1	Degradation (Half-life LacI)	[3,9]
deg_TetR	0.023 min^-1	Degradation (Half-life TetR)	[4,9,43]
deg_CIλ	0.0011 min^-1	Degradation (Half-life CIλ)	[4,9,43]
deg_Toxin	0.0025 min^-1	Degradation (Half-life Toxin)	[5]
A	6.02e²³	Avogadro’s number	[9]
V	1.5e^-16 dm^-3	Cell volume	[9]
P	1900	Number of RNA polymerases	[9]
N_ns	500,000	Non-specific binding sites	[1,2,9]
K_b	1.38e^-23m²kg s^-	Boltzmann constant	[9]
T	298	Temperature in Kelvin	-
∆ε_pd^S	0.01 nM	Energy of the state, RNA polymerases	[1,2]
∆ε_d^TetR_LacI	100 pM	Energy of the state, TetR repressor	[4,6,9]
∆ε_d^TetR_Toxin	100 pM	Energy of the state, TetR repressor	[4,6,9]
∆ε_d^LacI_TetR	1 pM	Energy of the state, LacI repressor	[3,9,43]
∆ε_d^CIλ_TetR	1 pM	Energy of the state, CIλ repressor	[4,9]
∆ε_d^CIλ_Toxin	1 pM	Energy of the state, CIλ repressor	[4,9]
nTet_LacI	-	Repressor binding sites TetR binding PLacI	-
nTet_Toxin	-	Repressor binding sites TetR binding PToxin	-
nCIλ_Toxin	-	Repressor binding sites CIλ binding PToxin	-
nCIλ_TetR	-	Repressor binding sites CIλ binding PTetR	-
nLacI	-	Repressor binding sites Laci binding PTetR	-

Designation	Value	Description	Reference
model iJP962	-	Genomic-scale metabolic model	[6,18]
carbonsource	-	Fructose, Glucose, Alanine, Asparagine, Aspartate, L-Leucine, Serine, L-Threonine, L-Proline, D-Glutamate	[15]
glucuptake	4.43 mmol gDW^-1hr^-1	Carbon uptake rate	[1]
NGAM	3.96 mmol gDW^-1hr^-1	Non-growth maintenance factor	[6]
KS_rate	0.2 μM min^-1	Protein production rate	Promoter Design
TAT_rate	1 nM min^-1	Toxin/anti-toxin production rate	Promoter Design
Cell density	0.040 gDW l^-1	gDW/OD600=0.1	[20]
ncells	1e⁸ cells ml^-1	cells/OD600=0.1	[21]
Chitinase	0.2 μM min^-1	Chitinase production rate	[7] and Promoter Design
Pyoverdine	150 μmol gDW^-1hr^-1	Pyoverdine production rate	Experiment
DAPG	25 μmol gDW^-1hr^-1	DAPG production rate	[8]
DMDS	0.845 μmol gDW^-1hr^-1	DMDS production rate	[9]
Nplasmids	10 plasmids cell^-1	Number of plasmids per cell (low copy number)	[23]
plasmidTD	0.3 h^-1	Time needed for at least a doubling of the number of plasmids per cell	[1]
TDrhizo	3 hr	Doubling time P. putida in the rhizosphere	[1]

Designation	Value	Description	Reference
K_{prod RNA_LacI}	0.75 µM min^-1	LacI production rate	[9,11]
K_{prod RNA_TetR}	0.75 µM min^-1	TetR production rate	[9,11]
K_{prod RNA_CIλ}	0.75 µM min^-1	CIλ production rate	[9,11]
K_{prod_RNAtoxin}	0.75 µM min^-1	Toxin production rate	[9,11]
deg_ToxinRNA	0.178 min^-1	Degradation (Half-life Toxin RNA)	[5]
deg_RNA	0.82 min^-1	Degradation (Half-life RNA)	[45]
deg_LacI	0.034 min^-1	Degradation (Half-life LacI)	[3,44]
deg_TetR	0.023 min^-1	Degradation (Half-life TetR)	[43]
deg_CIλ	0.011 min^-1	Degradation (Half-life CIλ)
deg_Toxin	0.000155 min^-1	Degradation (Half-life Toxin)	[38]
deg_Anti-Toxin	0.0011 min^-1	Degradation (Half-life Anti-Toxin)	[38]
Trans_LacI	0.12 µM min^-1	Translation Rate LacI	[9,45]
Trans_TetR	0.08 µM min^-1	Translation Rate TetR	[9,45]
Trans_CIλ	0.12 µM min^-1	Translation Rate CIλ	[9,45]
Trans_Zeta	0.12 µM min^-1	Translation Rate Zeta	[9,45]
K_TetR	4	Dissociation constant (TetR)	[4,6,9,43]
K_LacI	1.75	Dissociation constant (LacI)	[3,9,11]
K_CIλ	4	Dissociation constant (CIλ)	[4,43]
N_TetR	3	Hill constant(TetR)	[1,2,9,10]
N_LacI	2	Hill constant(LacI)	[3,9,43]
K_CIλ	2	Hill constant(CIλ)	[4]
α	1 nM min^-1	Free Toxin-Antitoxin production	[38]
σ	10	Antitoxin factor	[38]
μ_max	0.0153-0.0038	Maximum cell growth rate	Metabolic Model
KP₁	1000 nM	Dissociation constant(Toxin Repression)	[38]
KP₂	10 nM	Dissociation constant(antiToxin Repression)	[38]
KT₁	10	Dissociation constant(Growth inhibition)	[38]
KT₂	100	Dissociation constant(Translation inhibition)	[38]
n	6	Half of the total number of Toxin-Antitoxin systems	[38]
p	2	scaling factor	[38]
KH	1000 nM	Dissociation constant(Antitoxin)	[38]

Team:Wageningen UR/project/model

From 2014.igem.org

Model

Appendix

Modeling Overview

Kill-switch promoter design

Introduction

Theory and experimental design

Statistical Mechanics

Modeling Biological Systems using Statistical Mechanics

Model Equations

Software used

All models were made in Python (Python software foundation, Delaware, US) version 2.71 for Mac OS X version 10.5.8. Solving of ODE systems was done using the standard soln function in Python

Results

System Behaviour and Optimal Repressor Binding Site Configuration

Statistical Mechanics in Practice

System Cost

Introduction

Theory and experimental design

Extending the model

Programs and toolboxes

Flux Balance Analysis

Media composition

Results

Oxygen necessity in the soil

Parameter analysis results

System Performance

Introduction

Theory and experimental design

Stochastic Modeling and Toxin-Antitoxin Systems

Toxin-Antitoxin Systems

Model Equations

Model Sensitivity

Software used

All models were made in Python (Python software foundation, Delaware, US) version 2.71 for Mac OS X version 10.5.8. Solving of ODE systems was done using the standard soln function in Python

Results

The stability and performance of the Biological Control Agent over Cell Division

Conclusion

Appendix

Table of parameters

Promoter Design

System Cost

System Performance

References

Promoter Design

System Cost

System Design