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Wageningen UR iGEM 2014

Modeling Overview

Three aspects of the BananaGuard system have been modeled. (I) Accurate balancing of the different promoters elements in the the kill switch is required in order to maintain a bi-stable system. Using statistical mechanics an estimation for the optimal promoter configuration was made, balancing the amount of different repressors with number of repressor binding sites and their respective position on the promoter. The obtained results have inspired the design of promoters for the kill-switch. Having designed an optimal system its performance in the soil can be estimated in order to gain insight in its functionality. From a modeling perspective the most interesting questions that can be answered are: the metabolic price that has to be paid for the introduced genetic circuit and the stability and performance of the genetic circuit. (II) A genome scale metabolic model was extended outlining and answering the cost query and subsequently the biological control agents ability to not be outcompeted in the soil by other rizhosphere populating micro-organisms. (III) A stochastic model was made in order to quantify and characterize the stability and performance of the introduced genetic circuit. The model takes into account rates predicted by the metabolic model, cell division, the anti-gene transfer toxin antitoxin systems effect on cell growth and the kill-switch.

Schematic overview of the integration between the modelers themselves and the wetlab experimentalists

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].

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 [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 [3].

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].

This means that case specific designs cannot be modeled accurately but the likelihood of designing a functioning system can be estimated. 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 repressor binding site combinations can be determined (Figure 1). Statistical mechanics is an ideal tool because it can incorporate both repressor binding strength and binding sites. Learn more about the statistical mechanics approach or view the results.

**Figure 1.** The figure shows the kill-switch with arbitrarily chosen repressor binding sites. (I) Altering the position of the TetR repressor binding sites change the effectiveness; a large space of basepairs between the binding site and the LacI will result in weaker repression [8]. (II) Adding a LacI repressor binding site will increase the chance that LacI binds to the promoter region.

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, 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.

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 chance that a RNA polymerase binds to the promoter of interest a statistical mechanics model has two microstate outcomes:

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

(II) A state where the promoter of the gene of interest is occupied and the remaining polymerases are distributed among the non-specific binding sites (specific binding)6. In order to determine the probability that either state will be realized we need to know the total number of possible micro-sates and the chance that a particular microstate will be achieved. For the non-specific binding by RNA polymerases combinatorics can be used 5,6.

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

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:

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

The Boltzmann weight describes difference between strength of specific and non-specific binding. 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.

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

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.

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

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

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

In this form a fold change of mRNA expression can be correlated to the chance that a polymerase binds. If the situation is thus N_ns/P e^{(-(∆ε_pd^S)/(K_bT))} = 0, that the chance of an RNA polymerase binding is equal one, there is no change in expression (i.e. the fold change in expression remains one). 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 (fig.1). The effect of repressors can be taken up in equation (1.4) by adding the microstate ‘repressor protein bound’ as a regulatory region to the total statistical weight. 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 as equation (1.5) takes the form of a Hill function, the expression fold change is dependent on the number of repressors.

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

In equation (1.5) R is the number of repressor proteins and ∆ε_d^r the specific binding energy of the repressor proteins, the same combinatorics rules apply. 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 multiplied7. Equation (1.5) can now be incorporated into the model describing the kill-switch by tying production rates to the fold change of expression (K_prod x Fold change), i.e. the production rate will decrease proportionally to changes in expression.

Model Equations

Ordinary differential equations describe the genetic circuit (fig.4), three of which use a statistical mechanics based Hill function. The deg_protein and Dil_div terms represent the half-life dilution rate of the proteins. Because equation (2.3) (the CIλ production) relies on an external input and is not dependent on any of the other equations, the ODE uses Michaelis-Menten kinetics to simulate a Rhamnose input. In every simulation rhamnose is inputted at t = 10 hours

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

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

$\frac{d C I λ}{d t} = K_{p r o d_{λ C I}} \frac{[R H A]}{[R H A] + K_{d}} - {d e g}_{λ C I} C I λ - {D i l}_{d i v}$

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

Results

System Behaviour and Optimal Repressor Binding Site Configuration

Before the optimal configuration of repressor binding sites could be determined, the behavior of the system had to be assigned scores representing a functioning or non-functioning toggle switch (see Parameters). The scores range from zero to two, three different behaviors could be identified:

2: The system performs to design, after a rhamnose input the toggle switch changes state

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

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

In total eight different configurations of repressor binding sites that were tested. Two, F and H, were found to be moderately robust towards variations in binding strengths and production rates.

**Figure 1.** Color maps indicating functioning and non-functioning systems. Each letter represents different repressor binding site configurations. Each small square within the colour maps represents a score for a simulation of the system with a unique set of parameters. The colours correspond to the previously given description

Binding Sites	A	B	C	D	E	F	G	H
*TetR Promoter_LacI*	1	1	1	1	2	2	2	2
*TetR Promoter_GFP*	1	1	2	2	1	1	2	2
*CI Promoter_GFP*	2	2	1	1	2	2	1	1
*CI Promoter_TetR*	2	1	2	1	2	1	2	1
*LacI Promoter_TetR*	1	2	1	2	1	2	1	2

The result of the best scoring configuration H has been magnified (figure 2). 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 production rates in micro-molar per minute, 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.

Statistical Mechanics in Practice

Statistical mechanics allows you to 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. In contrast to Michaelis-Menten kinetics, statistical mechanics offers more freedom studying the behavior of differently designed promoters. Should Michaelis-Menten kinetics have been used the dissociation constant of a protein to DNA would need to be experimentally determined for different promoter configurations in order to obtain an indication of system behavior, a ‘parameter dependent’ method of modeling. Statistical mechanics is not bound by experimentation in making a rough predictions. Though the model still needs parameters e.g. number of RNA polymerases in a cell, number of non-specific binding sites, these are accessible without a need for experimentation1. The model predicts the only functioning repressor binding site configurations to be those where LacI and TetR are placed pairwise on the promoter, a result that is confirmed in the literature2. This means the statistical mechanics approaches can estimate which repressor binding site configuration is most likely to work. The information provided by this model has inspired the design of the promoters for the kill-switch. The next modeling step would be to assess the Cost and Performance of our entire system. This will give insight in the overall capacity and behavior of our biological control agent in the soil.

System Cost

Introduction

The BananaGuard system is designed with the assumption that our engineered Pseudomonas putida is not outcompeted by other bacteria and fungi found in the rhizosphere of banana roots[1,2]. However, our system consists of multiple constitutively expressed genes controlling the kill switch and toxin-antitoxin system. Moreover, when the system is switched to the active state by sensing fusaric acid, Pseudomonas putida produces multiple antifungal enzymes and compounds. The integrated synthetic pathways use metabolic resources that would otherwise be dedicated to cellular maintenance or growth. Thereby, the synthetic pathways reduce the bacteria’s potential to sustain themselves in the rhizosphere. Therefore, we wondered to what degree P. putida metabolism was affected by our system and, if need be, how the system could be changed to lessen the impact. To approach this problem a genome-scale metabolic model (GSMM) is used[3]. Our new GSMM predicts the interaction of our integrated plasmids with genes originating from endogenous pathways. With this indication of the metabolic stress we get an insight in the metabolic burden caused by the integration of our plasmids and whether the engineered Pseudomonas putida strain is losing the capability to compete with other rhizosphere-populating microorganisms in the resting state.

Theory and experimental design

In order to solve our problem 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
contains the most reactions that are necessary for our system

For this reason a comparison was made between various GSMM for P. putida. There are multiple genome-scale metabolic models available for Pseudomonas Putida but only two of them are up-to-date, published and verified: iJP962 and iJN746[4,5]. For the necessary work to model the metabolic stress in Pseudomonas Putida, the best starting model needs to be chosen which has the most potential for accurately modeling the metabolic stress of our system.

Genomic Model	iJP962	iJN746
Reactions	1070	957
Metabolites	959	685
Compartments	1	2
Focus of the model (based on metabolites and reactions)	This model has a broad spectrum of reactions in every class, but is lacking the in-depth reactions.	This model has a focus on the production of interesting compounds for use in biotechnology (aromatic compounds, alkenes, alkanoates).

Taking the above arguments in consideration, model iJP962 is most suitable for modeling our system. Even though the reactions and metabolites in iJN747 are divided in 2 compartments this is not an advantage. Many of the reactions are over the internal membrane and not functional for the core intermediate metabolites. On top of that iJN747 has a focus on compounds that are in a different class of metabolites then the toxins we chose for our system[5]. Therefore iJP962 is more representative of the real system due to more reactions and more metabolites in a broader spectrum. The primary category of concern for toxin production is the amino acid metabolism and iJP962 has the most reactions in this category[3,4,6]. Both models did not include all the specific reactions, metabolites and genes that are necessary for modeling the toxin metabolites. These reactions were added manually.

Metabolic modeling

Traditionally, metabolic models only predict metabolic fluxes. Although this is directly applicable to the metabolites that we want to produces, such as DMDS, this does not suffice to answer our questions regarding the impact of the whole system. Therefore, we added the production of all the proteins of interest as if they are metabolites. For this the protein sequences are used to determine the correct stoichiometric ratio of amino acids[7]. The necessary energy and other side-substrate/products are in the same ratio as described for production of proteins for the biomass reaction in the GSMM. In the model Promoter Design the protein production rate is estimated.

As our metabolites of interest (DAPG, DMDS, DMTS) were not yet included in the GSMM, we had to deduce their production pathways based on databases and literature information[8,9,10,11](link notebook with all reactions). Furthermore, we took special care that all added pathways are stoichiometrically balanced, which is essential for proper and reliable integration in a GSMM. To calculate the metabolic stress as accurately as possible, the correct constraints have to be set in the model to be certain that all the calculated production rates are biological relevant. To find which constraints are biological relevant in vivo data validation is important. Using this information an estimation can be made about the production range of the toxins which are relevant for P. putida.

Programs and toolboxes

Matlab[12] was used in combination with the Cobra toolbox[13] for handling the model. The Cobra toolbox is designed for working with genome-scale metabolic models. To solve the flux balance analysis a linear equation solver is needed. Gurobi[14] was chosen because of its compatibility with Matlab and the Cobra toolbox.

Flux Balance Analysis

Flux Balance Analysis (FBA) is a framework for investigating the theoretical capabilities of a stoichiometric metabolic model S and can be solved by using a linear solver program. This is possible because of linear objective function and linear constraints[4,6]. FBA is based on the following assumptions to ensure its linearity:

steady state: The fluxes are considered to have attained a static equilibrium value and do not change through time.
Thermodynamic constraints: (ir)reversibility of reactions
No enzyme saturation: The maximum flux for every reaction is not dependent on the concentration of the enzymes used for that reaction.
There is no accumulation of intermediate metabolites in the model.

The FBA method uses a representation of the metabolic reaction network in the form of a stoichiometry matrix (S) where :

Each column corresponds to a reaction R_i
Each row corresponds to a metabolite C_j

The definition of S is :

S_{i, j} = k s . t . \{\begin{matrix} k < 0 \to C_{j} i s a s u b s t r a t e o f r e a c t i o n R_{i} w i t h s t o i c h i o m e t r y k \\ k > 0 \to C_{j} i s a p r o d u c t o f r e a c t i o n R_{i} w i t h s t o i c h i o m e t r y k \\ k = 0 \to C_{j} d o e s n o t p a r t i c i p a t e i n r e a c t i o n R_{i} \end{matrix}

The FBA problem is then formulated as a maximization or minimization problem under the determined constraints:

\{\begin{matrix} M a x i m i z e c^{T} v \\ S u b j e c t t o S v = 0 \\ L o w e r b o u n d \leq v \leq U p p e r b o u n d \end{matrix}

Where :

v is the vector of unknown reaction fluxes
c is a vector of constants defining the objective function
S is the stoichiometry matrix
lowerbound and upperbound are vectors of constraints (minimal and maximal flux values for each reaction)

Media composition

For metabolic modeling the constraints of your exchange reactions are important for the results. With these exchange reactions you can dictate what’s available in the model, you need arbitrary bounds as actual uptake rates are unknown. This also enforces the later decision to not include all root components but only the most prevalent. You don’t want that rare compounds have a huge impact on the model predictions. This does imply that the model predictions could be slightly negative: with including the other available compounds the system may work even better.

In order to set the medium constraints of the models all lower and upper bounds are first set to the value of 0 and 1000 arbitrary units, respectively. Then one by one the lower and upper limit of the reaction bounds are set to -1000 for one reaction at a time to your composition of interest. If the lower bound is set to a value higher than zero a minimum flux of that value is forced in a FBA. To get the most realistic approximation for our engineered P. putida in the soil the M9 media composition was used with carbon as limiting nutrient. The carbon source is based on the exudates of banana roots and consists of 34% sucrose and 66% divided over multiple amino acids[15]. We have neglected other secondary metabolites in banana roots exudates because they are exudated in relatively small amounts[16].

Results

To check if BananaGuard is still viable in its resting and active state, a comparison is necessary between the wild type P. putida and BananaGuard. If the growth rate of BananaGuard approximates the growth rate of wild type P. putida in the resting state it is safe to assume that the disadvantage of the introduced synthetic pathway does not have a huge impact on competitiveness with other rhizosphere-populating microorganisms. This means that BananaGuard should be able to populate the rhizosphere of the banana roots. When looking at the active state we are not interested in the capability to compete with other microorganisms but the capability to produce all the anti-fungals with a constant rate around the roots, without overloading a metabolic pathway and causing shortage of intermediates.

P. putida is able to grow in the rhizosphere with a doubling time of 3 hours resulting in a growth rate of 0.23 h^-1[1]. Because of the linearity of the growth rate and carbon uptake rate in this model we expect that the carbon uptake rate is ≈4.5 mmol gDW^-1hr^-1 in the rhizosphere of the banana roots. Figure 1 depicts how the growth rate of BananaGuard compares to the growth rate of the wild type P. putida depending on the carbon uptake rate of the organism in the resting state and active state of the kill switch. To get the best case scenario glucose was chosen as carbon source reference, because of the efficient degradation of glucose. BananaGuard is still able to grow at almost max growth rate (>99%) compared to the wild type in the resting state. We can conclude out of this that BananaGuard is not getting outcompeted by other rhizosphere-populating microorganisms because of the synthetic pathway. When the kill switch is activated by fusaric acid BananaGuard is still able to growth with 50-70% of the maximum growth of the wild type (Figure 1). This indicates that metabolic stress is not a bottleneck for the production of anti-fungals in our fusaric acid activated system.

To get a more realistic result for the soil in banana farms the carbon source is changed to have the same ratio of carbohydrates and amino acids as found in the exudate's of banana roots[15]. This new carbon composition causes a reduced growth rate because of less efficient degradation of these compounds. In figure 1 the relative growth rate is shown for the resting and active state of the kill switch with the new carbon source. These changes in the carbon composition do not change our conclusion previously made. The range of the relative growth rate has changed from 50-70% to 45-60% indicating it still has the capacity to grow and maintain the constant toxin production with one condition that BananaGuard needs to stay close to the roots to maintain its carbon uptake rate.

Figure 2: The relative growth rate compared to the wild type *P. putida* for different carbon uptake rates. The realistic solution is with the banana exudates as carbon source and the other sulution is with glucose as reference. The expected carbon uptake rate of *P. putida* in the rhizosphere is indicated with transparent red.

Oxygen necessity in the soil

P. putida is a strict aerobic bacteria[2]. This means that without oxygen available the bacteria cannot grow. To inhibit the growth of the Fusarium oxysporum our engineered Pseudomonad must grow in the rhizosphere of the banana roots. If there is not enough oxygen present in the rhizosphere it might not be possible anymore to sustain anti-fungals production or other anaerobic bacteria can outcompete our P. putida. In figure 2 the growth rate in the active state is plotted as a function of the oxygen and carbon availability. If there is little oxygen available the growth rate drops and so will the production of the toxins. However the depletion of oxygen should not be a problem because there is a layer of oxygen present closely to the roots[17]. This supports the task of our P. putida to protect the banana roots from F. oxysporum invasion and will contain our engineered P. putida close to the banana roots.

Parameter analysis results

Every parameter in this metabolic model is an estimation based on literature growth experiments except for the pyoverdin production rate. This parameter has been determined from growth experiments with P. putida (Link notebook walter pyoverdin) . So every parameter is biological relevant, but because different studies report different values a mean has to be taken between the different data points. This means that the parameter can fluctuate in a small range. To compensate this a parameter analysis has been performed. Every parameter that could have a significant impact on the results has been checked over a range, dependent on the accuracy of the calculated parameter. The same analysis has been performed for all the fluctuations in the parameter set combined. This analysis shows the possible fluctuations of the modeled system and which parameters has greatest impact on the system.

System performance over cell division

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 toggle-switch in the presence of fluctuating protein concentrations and the activation of the kill-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 proteins1. 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 antifungal producing LacI state (Leakiness). A state that is vital for our biological control agent to work but only when induced in the presence of Fusarium 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 instability and characterize the biological control agents’ performance in the presence of Fusarium 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 1.** 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 the biological control agent is defined by its ability to destroy fusarium upon detection, its ability to compete with other micro-organisms and the stability of the genetic circuit. The ability to destroy Fusarium 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 1)

**Figure 1.** A schematic representation of the reengineered *P.putida*.(I) Noise can destabilize bistable systems, causing the kill-switch to change state. **(II)** A leaky promoter can destabalize the toggleswitch by weighing it to the LacI state. **(III)** the induction of the genetic circuit by fusaric acid determines the performance of *P.putida* as a biological control agent.

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 systems2. 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 our re-engineered Pseudomonas putida strain changes from the dormant state to the anti-fungal production state without an external input, the system would not function. The following model therefore aims to quantify the instability 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.

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 promoters3. 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 will weigh the toggle-switch in favour of the anti-fungal 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.

Induction (III) ties into the performance aspect of our biological control. The biological control agent has to perform in the presence of Fusarium oxysporum. It is imperative that the Kill-switch system is activated in 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 anti-fungal producing state.

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. 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 assess the population dynamics of our biological control agent6.

Model Equations

The model consists of eleven differential equations. Equations 1-6 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.

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

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

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

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

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

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

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. The 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.

Equation 7 describes the production of toxin RNA as part of the kill-switch similar to equations 1-6. Equations 8-9 describe the levels of free toxin and antitoxin through synthesis (term I), active degradation (term II), and stochastic fluctuations (term III). The first part in the denominator of the synthesis term describe the impact of protein degradation due to free toxin levels. The second part describes the auto repression of protein synthesis by the various complexes6.

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

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

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

Equation 10 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.

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

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

The last equation describes the concentration of fusaric acid in the cell. Not much is known about fusaric acid dynamics besides its quantities are measured 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 infection5. 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 biological control agent most likely found in nature.

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

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 the kills-witch for this parameter set. The values for the toxin-toxin papers were published by Rick A. Fasani and Michael A. Savageau (2013).

Table

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 1).

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 toggle 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 toggle 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. The effect of just noise on the stability of the biological control agent has been neglected because its effect was statistically insignificant.

Figure 2. 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 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. 20000 simulation were run

(B) For a Maximum growth rate of 180 minutes the stability of the kill-switch 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: 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. The same can be observed if the maximum growth rate is adjusted to the 180 minutes predicted by the metabolic model. Because the population of cells averaged a significantly higher growth rate the percentage of unstable kill-switches increased proportionally to, once again confirming increased instability for lengthy cell divisions cycles. 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.

Figure 3. 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: The population dynamics for an induced system varies slightly. The same logic applies with respect to leaky toxins, the timespans 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.
Overall the results provided by the stochastic model gives an initial indication of the system’s efficiency by quantifying its stability and performance. It gives insight into the behaviors the system might display in the soil. Finally it has shown that the biological control agent though potentially slowed down by the toxin anti-toxin system can work.

Appendix

Table of parameters

Metabolic modeling

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 [10,11,22]

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]

Dynamic modeling

References

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< i> Escherichia coli. Gene, 18(3), 335-341.

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@@ Line 573: / Line 573: @@
                          <h2 id="results3">Results</h2>
- <P>
+<P>
 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.<br/><br/>
 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 1).
 </P>
                          <figure>
                              <img src="https://static.igem.org/mediawiki/2014/thumb/9/96/Wagenigen_UR_model_bob_A.PNG/800px-Wagenigen_UR_model_bob_A.PNG" width="100%">
-                             <figcaption><b>Figure 2.</b> <b>A</b> The plot shows the impact of a normally distributed noise term over a range of 120 to 240 nM for the toggle switch proteins LacI and TetR. Over the course of 10000 cell division cycles per point, 5% of the toggle-switches appears to become unstable given a noise term of 200 nM and a 45 minute division time. Should the division time be doubled to 90 minutes less than half a percent becomes unstable. <b>B</b> The plot shows The impact of different  ΛCI protein basal production rates on toggle-switch stability over the course of 10000 cell division cycles per point. The noise term for the toggle-switch proteins LacI and TetR is 200nM.  This corresponds to 5% and 0.5% instability due to noise (fig. 1A). The plots show a linear increase in instability for increasing LamdaCI basal production rates given a cell division time of 45 minutes. The increase in instability is a lot higher with the. <b>C</b> The plot shows the performance levels of the kill-switch upon induction by Fusaric acid over the course of 10000 cell division cycles per point. Simulated with four different Fusaric acid inputs and four different growth rates. Testing solely the performance no leakiness was taken into account, a noise term of 200nM was added for the toggle-switch proteins LacI and TetR. The figure show the influx of Fusaric acid is not limiting to the performance of the kill-switch. The repeated doubling of the maximum growth rate doubles performance..   </figcaption>
+                             <figcaption><b>Figure 1.Plots showing the effect of noise and leakiness on the stability of the kill-switch ans the performance of the killswitch upon induction by fusric acid. Every plot point represent a percentage of unstable (leakyness and noise) or functioning (induction) kill-switches over 10000 simulations.
+							<br/><br/>
+							<b>(A)</b> 	The plot shows the impact of a normally distributed noise term over a range of 120 to 240 nM/min for the toggle switch proteins LacI and TetR. 5% of the toggle-switches appears to become unstable given a noise term of 200 nM/min and a 45 minute division time. Should the division time be doubled to 90 minutes less than half a percent becomes unstable.
+							<br/><br/>
+							<b>(B</b> The plot shows The impact of different  CIλ protein basal production rates on
+							toggle-switch stability. The noise term for the toggle-switch proteins LacI and TetR is 200nM.  This corresponds to 5% and 0.5% instability due to noise (figure. 1A). The plots show a linear increase in instability for increasing LamdaCI basal production rates given a cell division time of 45 minutes. Proportionally, the increase in instability is a lot higher given a doubling in cell division time umax to 90 minutes. This can be atributed to a lower dilution rate allowing for a build-up of CIλ increasing instability.
+							<br/><br/>
+							<b>C</b> The plot shows the performance levels of the kill-switch upon induction by Fusaric acid. Simulated with four different Fusaric acid inputs and four different growth rates. Testing solely the performance no leakiness was taken into account, a noise term of 200nM was added for the toggle-switch proteins LacI and TetR. The figure show the influx of Fusaric acid is not limiting to the performance of the kill-switch. The repeated doubling of the maximum growth rate doubles performance. Indicating that the build-up and therefore production rate of CIλ is limiting to the functioning of the kill-switch   </figcaption>
                          </figure>
+                        <p></p>
 <P>
 <b>Noise, A:</b> 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 toggle 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 toggle switches that have changed state. <br/><br/>
@@ Line 615: / Line 628: @@
   </figcaption>
                          </figure>
+                        <p></p>
 <P>
@@ Line 624: / Line 638: @@
                           <figure>
                              <img src="https://static.igem.org/mediawiki/2014/0/00/Wageningen_UR_model_bob_C.png" width="100%">
-                             <figcaption><b>Figure 2.</b> <b>A</b>The figures show the stability given a leaky promoter and 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 given the investigative perspective and a specific cell division time, red bars indicate non-functioning systems. For both the simulations determining promoter leakiness and system induction two maximum growth rates uMax were investigated. Both the maximum Pseudomonas putida growth rate of 45 minutes and  </figcaption>
+                             <figcaption><b>Figure 3.</b> <b>A</b>The 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.<br/><br/>
+							(A)  For a maximum growth rate of 45 minutes 36% of the kill-switches activate, longer division times activate the cells more effectively.<br/><br/> (B) For a maximum growth rate of 180 minutes 98% of the kill-switches activate, longer division times activate the cells more effectively. </figcaption>
                          </figure>
+                        <p></p>
 <P>
 <b>Induction:</b> The population dynamics for an induced system varies slightly. The same logic applies with respect to leaky toxins, the timespans 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.

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	[10,11,22]
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]

Team:Wageningen UR/project/model

From 2014.igem.org

Revision as of 09:31, 16 October 2014

Model

Appendix

Modeling Overview

Kill-switch promoter design

Introduction

Theory and experimental design

Statistical Mechanics

Modeling Biological Systems using Statistical Mechanics

Model Equations

Results

System Behaviour and Optimal Repressor Binding Site Configuration

Statistical Mechanics in Practice

System Cost

Introduction

Theory and experimental design

Metabolic modeling

Programs and toolboxes

Flux Balance Analysis

Media composition

Results

Oxygen necessity in the soil

Parameter analysis results

System performance over cell division

Introduction

Theory and experimental design

Stochastic Modeling and Toxin-Antitoxin Systems

Toxin-Antitoxin Systems

Model Equations

Model Sensitivity

Results

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

Appendix

Table of parameters

Metabolic modeling

Dynamic modeling

References