Section1. Flow of Gene Expression to PET Degradation

Overview 1.1. Section1 (Part1)

Overview 1.2. Section1 (Part2-Part4)

Part1. Gene Circuits Model

Description of Gene Circuits

In this part, we developed a model of the Electro Count Biosafety Module (ECB Module) to evaluate the behavior of the electrical response gene circuit, drawing inspiration from the initial lab experiments. Based on insights from those experiments, the dry lab model was used to estimate optimal conditions. This includes the activation of the electrical-responsive promoter, pSoxS, site-specific recombination of Cre-loxP, the expression of BIND-PETase and MazF from the electrical stimulation. These estimated conditions were then applied to the lab experiments in an attempt to validate the promoter’s function. However, due to challenging experimental conditions, the lab experiments were unable to demonstrate its operation. Despite this, our model successfully validated the ECB Module system, which we are first to model this scheme as an iGEM team, offering valuable insights for optimizing future lab experiments.

Model Structure

Fig. 11.2.1.1. Gene Circuits Model

The outlined steps of the ECB Module are shown below. (Fig. 11.2.1.1)

Step 1: Add electrical stimulation. Reduced Pyocyanin ( $Pyo_{red}$ ) transforms into the oxidized form ( $Pyo_{ox}$ ).
Step 2: $Pyo_{ox}$ oxidizes SoxR protein. The oxidized SoxR ( $SoxR_{ox}$ ) binds to the two sites of pSoxS, activating the promoter. To note, the expression of reduced SoxR ( $SoxR_{red}$ ) is homeostatic.
Step 3: The first pSoxS promotes the expression of Cre Recombinase in the downstream.
Step 4: Expressed Cre binds to the loxP sequence found in the two genes. This activates the Cre/loxP recombination system, cutting out the sequence surrounded by loxP.
Step 5: The gene with the constitutive promoter expresses the downstream genes. We implemented a fusion protein called “BIND-PETase”, including both CsgA and PETase. In the meantime, reduction voltage reduces SoxR, restricting the activity of the pSoxS promoter.
Step 6: Second electrical stimulation is added to kill the E.coli itself. In the other gene with loxP sites, SoxR is oxidized, thus the pSoxS promoter activates to express MazF. This sterilizes the E.coli, preventing leakage.

1. Pyocyanin and SoxR

We modeled the initial reaction of the ECB Module by the redox reaction of Pyocyanin in Step 1 and Step 2. A previous project conducted by iGEM Imperial College London (ICL, 2018)^[1] established a similar model, which formulated the redox reaction between Pyocyanin and SoxR, by using phenomenological functions to fit reaction constants. However, the model did not provide insights into the time variation. Therefore, we created an Ordinary Differential Equation (ODE) to make theoretical predictions about how the reaction proceeds and its time dependence.

Fig. 11.2.1.2. Reaction scheme of Pyocyanin and SoxR^[2]

In the process of formulation, we made several assumptions.

Since Pyocyanin is hydrophobic^[4], we did not consider time lag from the flow of it.

\begin{align} \frac{d[Pyo_{ox}]}{dt}=&-k_{ox}[Pyo_{ox}][SoxR_{red}]\notag\\ &+k'_{ox}\exp{\left(\frac{(1-\alpha_{1})F\Delta\phi}{RT}\right)}[Pyo_{red}]\notag\\ &-k'_{red}\exp{\left(\frac{-\alpha_{1}F\Delta\phi}{RT}\right)}[Pyo_{ox}]\\ \frac{d[Pyo_{red}]}{dt}=&k_{ox}[Pyo_{ox}][SoxR_{red}]\notag\\ &-k'_{ox}\exp{\left(\frac{(1-\alpha_{1})F\Delta\phi}{RT}\right)}[Pyo_{red}]\notag\\ &+k'_{red}\exp{\left(\frac{-\alpha_{1}F\Delta\phi}{RT}\right)}[Pyo_{ox}]\\ \frac{d[SoxR_{ox}]}{dt}=&k_{ox}[Pyo_{ox}][SoxR_{red}]-deg_{SoxR_{ox}}[SoxR_{ox}]\\ \frac{d[SoxR_{red}]}{dt}=&rbs_{SoxR_{red}}[mRNA_{SoxR_{red}}]-k_{ox}[Pyo_{ox}][SoxR_{red}]\notag\\ &-deg_{SoxR_{red}}[SoxR_{red}]\\ \frac{d[mRNA_{SoxR_{red}}]}{dt}=&{N_{plasmid}}{P_{SoxR_{L}}}-deg_{mRNA_{SoxR_{red}}}{[mRNA_{SoxR_{red}}]}\\ {G}=&\frac{K_{red}^n}{K_{red}^n+{[SoxR_{red}]}^n} \cdot \frac{[SoxR_{ox}]^n}{K_{ox}^n+[SoxR_{ox}]^n}\cdot {P_{SoxR}} \end{align}

The equations were set under the balance of supply and consumption. For Equations 1 and 2, $\Delta \phi$ is a variable indicating the voltage to be applied, and on the first input, $\Delta \phi$ is larger than 0, which the quantity of supply exceeds that of the consumption by oxidation. Between the first and second input, $\Delta \phi$ is less than 0, and $Pyo_{ox}$ decreases as the reduction voltage is applied.

Equation 6 outputs the promoter robustness, where $SoxR_{ox}$ binds to the pSoxS promoter proactively and $SoxR_{red}$ binds inhibitively, with the addition of the hill equation. Wiki from ICL was referred to, however since the formulas were excessively simplified, we calculated from the standpoint of definition. $G$ is the function that controls the gene expression of the pSoxS promoter, revealing the potency of the promoter.

2. Cre Recombination

Step 3 and 4 of the ECB Module are related to Cre-Recombination. The Cre-recombination model was introduced by iGEM Utokyo (2022)^[5], creating the fundamental principles for modeling Cre-recombination. However, the equations were not written as an ODE. Therefore, we have created an ODE for the Cre Recombinase since its expression is regulated by the first pSoxS promoter.

After the pSoxS promoter is activated by Pyocyanin and SoxR, the downstream expresses Cre Recombinase and it cleaves the loxP-TT sequence upstream of the plasmid in which the target proteins, BIND-PETase (We call it BIP in equation,simulation for simplicity) and MazF, are introduced. This reaction can be divided into several stages by distinguishing how many and where Cre binds to the loxP-TT sequence. (Fig. 11.2.1.3)

Fig. 11.2.1.3. The Cre-recombination reaction scheme

\begin{align} \frac{d[mRNA_{Cre}]}{dt}=&G\cdot N_{plasmid}-deg_{mRNA_{Cre}} {[mRNA_{Cre}]}\\ \frac{d[S_0]}{dt}=&-k_{+1}[S_{0}][E]+k_{-1}[S_{1}]\\ \frac{d[S_1]}{dt}=&k_{+1}[S_{0}][E]-k_{-1}[S_{1}]-k_{+1}[S_{1}][E]+k_{-1}[S_{22}]\notag\\ &-k_{+2}[S_{1}][E]+k_{-2}[S_{21}]\\ \frac{d[S_{21}]}{dt}=&-k_{+1}[S_{21}][E]+k_{-1}[S_{3}]+k_{+2}[S_{1}]+k_{-2}[S_{21}]\\ \frac{d[S_{22}]}{dt}=&-k_{+2}[S_{22}][E]+k_{-2}[S_{3}]+k_{+1}[S_{1}][E]-k_{-1}[S_{22}]\\ \frac{d[S_3]}{dt}=&k_{+2}[S_{22}][E]-k_{-2}[S_{3}]+k_{+1}[S_{21}][E]-k_{-1}[S_{3}]\notag\\ &-k_{+2}[S_{3}][E]+k_{-2}[S_{4}]\\ \frac{d[S_{4}]}{dt}=&k_{+2}[S_{3}][E]-k_{-2}[S_{4}]-k_{+3}[S_{3}][E]+k_{-3}[I]\\ \frac{d[I]}{dt}=&k_{+3}[S_{3}][E]-k_{-3}[I]-k_{+4}[I]\\ \frac{d[P]}{dt}=&k_{+4}[I]\\ \frac{d[E]}{dt}=&k_{trans_{mRNA_{Cre}}}{[mRNA_{Cre}]}-k_{+1}[S_{0}][E]+k_{-1}[S_{1}]\notag\\ &+k_{+1}[S_{1}][E]+k_{-1}[S_{22}]-k_{+2}[S_1][E]+k_{-2}[S_{21}]\notag\\ &-k_{+1}[S_{21}][E]+k_{-1}[S_3]-k_{+2}[S_{22}][E]+k_{-2}[S_3]\notag\\ &-k_{+2}[S_{3}][E]+k_{-2}[S_{4}]-deg_{Cre}[E]\\ S&=[S_{0}]+[S_{1}]+[S_{21}]+[S_{22}]+[S_{3}]+[S_{4}]+[I]+[P]\\ \end{align}

Equation (17) shows the sum of total concentration of genes with loxP-TT sequences.The ratio $\frac{P}{S}$ of the concentration of the final cleaved sequence $[P]$ to $[S]$ indicates the progress of the Cre-recombination reaction. Similar to the variable $G$ for pSoxS, it is a variable illustrating control by the Cre-loxP system.

3. Downstream Gene Expression

Step 5 and 6 include models for downstream gene expression. In this model, we have created an ODE for its expression regulated by the Cre-loxP system. Several variables from 1. and 2. are incorporated, such as $G$ and $\frac{[P]}{S}$ , so that the expression is only done when the gene circuit functions properly. These steps become the final stage of the gene circuit model, thus the proper operation of this model is essential for the feasibility of utilizing the ECB Module.

Fig. 11.2.1.4. The reaction in 1st stimulation

\begin{align} \frac{d[mRNA_{BIP}]}{dt}&=k_{basal}+k_{max}\frac{[P]}{S}-deg_{mRNA_{BIP}}{[mRNA_{BIP}]}\\ \frac{d[BIP]}{dt}&=k_{trans_{BIP}}[mRNA_{BIP}]-deg_{BIP}[BIP]\\ \frac{d[mRNA_{mazF}]}{dt}&=k'_{basal}+G\cdot N_{plasmid}\frac{[P]}{S}-deg_{mRNA_{mazF}}{[mRNA_{mazF}]}\\ \frac{d[mazF]}{dt}&=k_{trans_{mazF}}[mRNA_{mazF}]-deg_{mazF}[mazF] \end{align}

There are three aspects that we considered to have a high influence on the mRNA levels in the downstream genes of the ECB Module^[6]. They are, the term showing leakage by being uncontrollable, the term which regulates correctly as assumed, and the term that exhibits autodegradation. As shown in Fig 11.2.1.4, the BIP gene is affected by the loxP-TT sequence, and MazF is affected by the loxP-TT sequence and the pSoxS promoter. To consider this, the control variables $G$ and $\frac{[P]}{S}$ exist in the second term of each mRNA reaction.

Abbreviations

Table. 11.2.3.1. List of Abbreviations and initial value of variables

Abbreviation	Description	Initial Value
$[Pyo_{ox}]$	Oxidized pyocyanin	$0.1$
$[Pyo_{red}]$	Reduced pyocyanin	$0.4$
$[SoxR_{ox}]$	Oxidized SoxR	$0.0$
$[SoxR_{red}]$	Reduced SoxR	$0.0$
$[mRNA_{SoxR_{red}}]$	mRNA of reduced SoxR	$0.0$
$G$	Strength of the pSoxS promoter	$0.0$
$[mRNA_{Cre}]$	mRNA of Cre	$0.0$
$[S_{0}]$	loxP sequence without Cre attached	$10.0$
$[S_{1}]$	loxP sequence with one Cre attached	$0.0$
$[S_{21}]$	loxP sequence with two Cre attached but separated	$0.0$
$[S_{22}]$	loxP sequence with two Cre attached close together	$0.0$
$[S_3]$	loxP sequence with three Cre attached	$0.0$
$[S_{4}]$	loxP sequence with four Cre attached	$0.0$
$[I]$	Circularized loxP sequence	$0.0$
$[P]$	Split loxP sequence	$0.0$
$[E]$	Cre Recombinase	$0.0$
$S$	All circularized loxP sequences	$10.0$
$mRNA_{BIP}$	mRNA of BIP	$0.0$
$BIP$	BIND-PETase protein	$0.0$
$mRNA_{mazF}$	mRNA of MazF	$0.0$
$mazF$	MazF protein	$0.0$

Parameters

Table. 11.2.3.2. List of Parameters

Name	Definition	value[unit]	Reference
$k_{ox}$	Conjugate reaction between oxidized Pyocyanin and reduced SoxR	$1.0 \times 10^{-3}$ [(mol･s) $^{-1}$ ]	Assumed:from the literature^[7], we can assume it as the value which is smaller than 1000
$k’_{ox}$	Rate constant of reaction of oxidation of Pyocyanin	$2.47\times10^{-8}$ [s $^{-1}$ ]	Calculated from experiment data.
$k’_{red}$	Rate constant of reaction of reduction of Pyocyanin	$1.0\times10^{-2}$ [s $^{-1}$ ]	Assumed
$\alpha_{1}$	Electric transfer coefficient of Pyocyanin	$0.5$ [-]	Assumed:from literature ^[8]
$F$	Faraday constant	$95495$ [C/mol]	^[9]
$R$	Gas constant	$8.3144$ [J･(K･mol) $^{-1}$ ]	^[10]
$T$	Absolute temperature	$310$ [K]	From experiment condition
$deg_{SoxR_{ox}}$	Natural degradation Rate of oxidized SoxR	$1.0\times10^{-6}$ [s $^{-1}$ ]	^[11] the average value of protein
$deg_{Sox_{red}}$	Natural degradation Rate of reduced SoxR	$1.0\times10^{-6}$ [s $^{-1}$ ]	^[11]: the average value of protein
$N_{plasmid}$	The plasmid copy number	$15$ [-]	From lab experiment conditions
$P_{SoxR_{L}}$	Promoter strength of Left side of psoxS promoter	$1.0$ [mol/s]	^[12]
$deg_{mRNA_{SoxR_{red}}}$	Natural degradation Rate of mRNA of reduced SoxR	$2.5\times 10^{-3}$ [s $^{-1}$ ]	^[13]
$K_{red}$	Dissociation constant of Reduced SoxR	$1.1 \times 10^{-10}$ [mol]	^[14]
$K_{ox}$	Dissociation constant of Reduced SoxR	$4.5 \times 10^{-10}$ [mol]	^[14]
$n$	Hill coefficient for SoxR	$1.6$ [-]	^[15]
$P_{SoxR}$	The strength of right hand of pSoxS promoter	$5.8 \times 10^{-3}$ [mol/s]	^[1]
$k_{+1}$	The rate constant of forward reaction of first Cre attaching to loxP sequence	$2.2 \times 10^{-1}$ [(mol･s) $^{-1}$ ]	^[5]
$k_{+2}$	The rate constant of forward reaction of second Cre attaching to loxP sequence	$0.66 \times 10^{-1}$ [(mol･s) $^{-1}$ ]	^[5]
$k_{+3}$	The rate constant of forward reaction of loxP sequence forming cyclic structure	$1.23 \times 10^{-3}$ [(mol･s) $^{-1}$ ]	^[5]
$k_{+4}$	The rate constant of forward reaction of cyclic loxP sequence cutting.	$2.94 \times 10^{-4}$ [(mol･s) $^{-1}$ ]	^[5]
$k_{-1}$	The rate constant of reverse of first Cre attaching to loxPsequence	$6.6 \times 10^{-2}$ [s $^{-1}$ ]	^[5]
$k_{-2}$	The rate constant of reverse reaction of second Cre attaching to loxP sequence	$4.8 \times 10^{-3}$ [s $^{-1}$ ]	^[5]
$k_{-3}$	The rate constant of reverse reaction of loxP sequence forming cyclic structure	$1.15 \times 10^{-1}$ [s $^{-1}$ ]	^[5]
$k_{trans_{mRNA_{Cre}}}$	The translation rate of mRNA of Cre	$0.183$ [s $^{-1}$ ]	^[1]: using the gfp’s value
$deg_{mRNA_{Cre}}$	The natural degradation rate of mRNA of Cre	$2.5 \times 10^{-3}$ [s $^{-1}$ ]	^[16]
$deg_{Cre}$	The natural degradation rate of Cre	$1.0 \times 10^{-10}$ [s $^{-1}$ ]	^[17]
$k_{basal}$	The leaky transcription rate of BIND-PETase	$1.0 \times 10^{-5}$ [mol/s]	Assumption
$k_{max}$	The theoretical maximum transcription rate	$10.0$ [mol/s]	^[12]
$deg_{mRNA_{BIP}}$	The natural degradation rate of BIND-PETase	$2.5 \times 10^{-3}$ [mol/s]	^[18]
$k_{trans_{BIP}}$	The translation rate of mRNA of BIND-PETase	$0.183$ [s $^{-1}$ ]	^[1]: using the gfp’s value
$deg_{BIP}$	The natural degradation rate of BIND-PETase	$1.0 \times 10^{-4}$ [mol/s]	^[16]
$k’_{basal}$	The leaky transcription rate of MazF	$1.0 \times 10^{-8}$ [mol/s]	Assumption
$deg_{mRNA_{mazF}}$	The natural degradation rate of mRNA of MazF	$2.5 \times 10^{-3}$ [mol/s]	^[19]
$k_{trans_{mazF}}$	The translation rate of MazF	$0.183$ [s $^{-1}$ ]	^[1]:using the value of gfp
$deg_{mazF}$	The natural degradation rate of MazF	$1.0 \times 10^{-4}$ [mol/s]	^[20]

Results

1. Pyocyanin and SoxR

Prior to the simulation, the parameter $k’_{ox}$ was estimated based on the lab experiments. It was necessary to measure this parameter since the electrical-responsive promoter is highly sensitive, and making assumptions about this parameter for Pyocyanin could result in inaccurate or misleading results. Therefore, we estimated the value of $k’_{ox}$ , and obtained $2.47 \times 10^{-8}$ [ $s^{-1}$ ].

In the simulation, $\Delta\phi=-2.2V$ was set until $100s$ to confirm that the promoter is driven only after the oxidation voltage is added, and $\Delta\phi=1.3V$ was set after $100s$ to confirm that the promoter is driven only after the voltage is added. As a result, as soon as the simulation started, $Pyo_{ox}$ , which was initially present because no oxidation voltage was applied, was consumed by the conjugation reaction and became almost zero.

Subsequently, when voltage was applied, electrical supply was provided and the concentration of the oxidized form increased. $SoxR_{ox}$ , which was almost unchanged, increased accordingly.

Then, the $[Pyo_{ox}]$ became a large fraction of all Pyocyanin, and the supply of $[Pyo_{red}]$ decreased steadily due to consumption by the conjugation reaction. By applying oxidation voltage, we observed an increase in $G$ , a function for the promoter strength, theoretically verifying the activation of the pSoxS promoter.

Fig. 11.2.1.5. The activated pSoxS promoter:Oxidized SoxR by Pyocyanin

The change was then gradual so that the amount consumed was balanced by the amount supplied by the reduction to the oxidized body due to voltage.
Then, the reduction to the oxidized form by voltage led to changes being gradual so that the amount supplied and consumed were in balance. As shown in the simulation results with simulation time:t = 10,000, the concentration of $[Pyo_{ox}],[Pyo_{red}],[SoxR_{ox}],[SoxR_{red}]$ converged into a fixed value.

Fig. 11.2.1.6. The equilibrium of $[Pyo_{ox}]$ and $[Pyo_{red}]$ at $t =10000s$

Fig. 11.2.1.7. The equilibrium of $[SoxR_{ox}]$ at $t =10000s$

Fig. 11.2.1.8. The equilibrium of $[SoxR_{red}]$ at $t =10000s$

2. Cre Recombination

The results show that $S_{0}$ , which is in the initial state, gradually decreased, while $P$ , which are excised by Cre, increased. These results confirm that the pSoxS promoter is activated by electrical stimulation, and thereby the Cre-loxP system behaves as expected. Also, we have confirmed that 50 seconds of casting oxidation voltage input is enough to activate pSoxS promoter and the Cre-recombination system.

Fig. 11.2.1.9. The concentration of loxP sequence in initial state

Fig. 11.2.1.10. The concentration of loxP sequence which was cut off.

3. Downstream Gene Expression

Out of all the ODEs that make up the ECB module, we made simulations except for MazF under square wave voltage. Note that MazF will be discussed in Part 5, since it is an important part of the kill switch for E. coli viability.

Fig. 11.2.1.11. The activated pSoxS promoter: $SoxR_{ox}$ by Pyocyanin

It has been confirmed that the expression of downstream genes was suppressed when no oxidation voltage was applied. This means that the electrical-responsive promoter functions properly, since the gene expression was controlled by electrical stimulations. From this, we now know they would be expressed if oxidation voltage is applied. The lab experiment aimed to express genes were done based on the knowledge of voltage inputs and initial conditions.

For further improvements, if the leakage expression levels, $k_{basal}$ or $k'_{basal}$ can be reduced, and the ECB Module would contribute further to biosafety. Here, we consider improvements from the viewpoint of a lab experiment. In this model, the plasmid leakage and the part controlled by the pSoxS promoter were considered separately. If we consider to artificially manipulate and reduce expression levels of RNA polymerase that transcribes the gene sequence, $k_{basal}$ and $k'_{basal}$ can be reduced in the above equation. We believe that it is possible to make them smaller than the values set in this model.

In total, Part 1 discusses the basic scheme of the ECB Module, and proves that the pSoxS promoter reacts to the electrical stimulation, as anticipated. The electrical stimulation led to the redox reaction of Pyocyanin and SoxR, Cre-recombination, and the ultimate downstream gene expression of BIND-PETase. For this, we are the first team in iGEM to model the behavior of the pSoxS promoter. Meanwhile, since we plan to use this module as a biosafety method, a second electrical stimulation is applied and will be discussed in Part 5.

Part2. Surface Display

Description of Surface Display

In this part, we modeled the process of extra-membrane transport of BIND-PETase (CsgA-PETase) expressed by the ECB Module and CsgA from the original genomic DNA of E.coli. In the lab experiment, the Congo Red Assay was performed to assess the quantity of the formation of Curli Fiber. However, it was not possible to directly measure the quantity of Curli Fiber, as there are issues mentioned in the lab experiment, such as detecting CsgA amyloids or Curli Fibers falling off. Therefore, in the dry lab, we have quantitatively examined the display of CsgA protein and other related proteins, as well as the Curli Fiber formation mentioned in Part 3.

Model Structure

Fig. 11.2.2.1. Overview: Formation of proteins and display of Curli fiber in E.coli

Most of the biological mechanisms of the Curli Fiber formation in E.coli are already known^[21]. As shown in Fig. 11.2.2.1, CsgA and CsgB are expressed starting from the expression of RpoS from E. coli and migrate to the outer membrane to form Curli fibers.

Model Construction

1. Protein Expression in the cytoplasm of E.coli

The starting point for Curli Fiber formation is the expression of RpoS, a sigma factor, in the cytoplasm. The expression of RpoS can be explained as follows.

\begin{align} \frac{d[RpoS]}{dt}=k_{transRpoS}-k_{degRpoS}[RpoS]\\ \end{align}

The expressed RpoS then acts on the csgDEFG operon to express CsgD, CsgE, CsgF, and CsgG. The respective expression levels can be written as shown.

\begin{align} \frac{d[D_c]}{dt}&=k_{transBG_c}[RpoS]-k_{degD_c}[D_c]\\ \frac{d[E_c]}{dt}&=k_{transBG_c}[RpoS]-k_{degE_c}[E_c]-k_{1E}[E_c]\\ \frac{d[F_c]}{dt}&=k_{transBG_c}[RpoS]-k_{degF_c}[F_c]-k_{1F}[F_c]\\ \frac{d[G_c]}{dt}&=k_{transBG_c}[RpoS]-k_{degG_c}[G_c]-k_{1G}[G_c]\\ \end{align}

Among these, CsgD in particular binds to the promoter of csgBAC operon and activates the transcription of CsgA, CsgB, and CsgC^[21]. Since CsgA is not only genome-derived but also BIND-PETase-derived as described in part 1, the total amount of CsgA can be expressed by adding the BIP terms.

\begin{align} \frac{d[A_c]}{dt}=&k_{transA_c}[D_i]-k_{degA_c}[A_c]-k_{1A}[A_c]\notag\\ &-k_{trans_{BIP}}[mRNA_{BIP}]+deg_{BIP}[BIP]\\ \frac{d[B_c]}{dt}=&k_{transBG_c}[D_c]-k_{degB_c}[B_c]-k_{1B}[B_c]\\ \frac{d[C_c]}{dt}=&k_{transBG_c}[D_c]-k_{degC_c}[C_c]-k_{1C}[C_c]\\ \end{align}

2. Translocation of expressed proteins to the Periplasm

Proteins expressed in 1. migrate to the outer membrane to form Curli Fiber at the outer membrane. The first membrane passing through is a translocon called secYEG. The process by which CsgA, CsgB, CsgC, CsgD, CsgE, CsgF and CsgG pass through this membrane is formulated as follows.

\begin{align} \frac{d[A_p]}{dt}&=k_{1A}[A_c]-k_{degA_p}[A_p]-k_{2A}[A_p]-k_{amyloid}[A_{p}]\\ \frac{d[B_p]}{dt}&=k_{1B}[B_c]-k_{degB_p}[B_p]-k_{2B}[B_p]\\ \frac{d[C_p]}{dt}&=k_{1C}[C_c]-k_{degC_p}[C_p]\\ %\frac{d[D_p]}{dt}&=k_{1D}[D_i]-k_{degD_p}[D_p]\\ \frac{d[E_p]}{dt}&=k_{1E}[E_c]-k_{degE_p}[E_p]-k_{on_{E}}[E_{p}]^9+k_{off_{E}}[E_{p_{9}}]\\ \frac{d[F_p]}{dt}&=k_{1F}[F_c]-k_{degF_p}[F_p]-k_{2F_{e}}[F_p]\\ \frac{d[G_p]}{dt}&=k_{1G}[G_c]-k_{degG_p}[G_p]-k_{on_{G}}[G_{p}]^9+k_{off_{G}}[G_{p_{9}}]\\ k_{amyloid}&=\frac1{1+\frac{[C_{p}]}{K_{i}}}\\ \end{align}

3. Transport of CsgA and B, with the support of CsgGEF

CsgA, B, C, E, F, and G in the periplasm have independent roles: CsgA is a component of the Curli Fiber itself, CsgB is a factor connecting CsgA to the plasma membrane, CsgC is a factor inhibiting amyloidogenesis of CsgA in the periplasm, CsgF is a factor localizing CsgB, and CsgE and CsgG function as transporters that bind to and transport CsgA and B out of the plasma membrane. CsgE and CsgG are known to bind to each other after forming a trimer, and two CsgF molecules bind to this complex, forming the CsgGEF complex^[22]^[23]. Here, we have formulated the formation of CsgGEF, as well as the transport of CsgA and B to the extracellular.

\begin{align} \frac{d[E_{p_{9}}]}{dt}&=k_{on_{E}}[E_{p}]^9-k_{off_{E}}[E_{p_{9}}]-k_{on_{GEF}}[E_{p_{9}}][F_{e}]^2[G_{p_{9}}]\\ &+k_{off_{GEF}}[GEF]-{k_{degEp9}}[E_{p9}]\notag\\ \frac{d[G_{p_{9}}]}{dt}&=k_{on_{G}}[G_{p}]^9-k_{off_{G}}[G_{p_{9}}]-k_{on_{GEF}}[E_{p_{9}}][F_{e}]^2[G_{p_{9}}]\\ &+k_{off_{GEF}}[GEF]-{k_{degGp9}}[G_{p9}]\notag\\ \frac{d[GEF]}{dt}&=k_{on_{GEF}}[E_{p_{9}}][F_{e}]^2[G_{p_{9}}]-k_{off_{GEF}}[GEF]-{k_{degGEF}}[GEF]\\ \frac{d[A_e]}{dt}&=k_{2A}[A_p]-{k_{degAe}}[A_{e}]\\ \frac{d[B_e]}{dt}&=k_{2B}[B_p]-{k_{degBe}}[B_{e}]\\ \frac{d[F_e]}{dt}&=k_{2F_{e}}[F_p]-k_{on_{GEF}}[E_{p_{9}}][F_{e}]^2[G_{p_{9}}]+k_{off_{GEF}}[GEF]-{k_{degFe}}[F_{e}]\\ \end{align}

Abbreviations

Table. 11.2.2.1. List of Abbreviations

Abbreviation	Description
$BIP$	BIND-PETase (CsgA + PETase)
$A_{c}$	Cytoplasmic CsgA
$B_{c}$	Cytoplasmic CsgB
$C_{c}$	Cytoplasmic CsgC
$D_{c}$	Cytoplasmic CsgD
$E_{c}$	Cytoplasmic CsgE
$F_{c}$	Cytoplasmic CsgF
$G_{c}$	Cytoplasmic CsgG
$A_{p}$	Periplasmic CsgA
$B_{p}$	Periplasmic CsgB
$C_{p}$	Periplasmic CsgC
$E_{p}$	Periplasmic CsgE
$F_{p}$	Periplasmic CsgF
$G_{p}$	Periplasmic CsgG
$E_{p9}$	Periplasmic CsgE (Nonamer)
$G_{p9}$	Periplasmic CsgG (Nonamer)
$GEF$	Complex of CsgGEF
$A_{e}$	Extracellular CsgA
$B_{e}$	Extracellular CsgB
$F_{e}$	Extracellular CsgF

Initial Conditions

Table. 11.2.2.2. List of Initial Conditions

Abbreviation	Description	Initial Value
$RpoS$	Cytoplasmic RpoS	$1.0 \times 10^{-6}$
$A_{c}$	Cytoplasmic CsgA	$0.0$
$B_{c}$	Cytoplasmic CsgB	$0.0$
$C_{c}$	Cytoplasmic CsgC	$0.0$
$D_{c}$	Cytoplasmic CsgD	$0.0$
$E_{c}$	Cytoplasmic CsgE	$0.0$
$F_{c}$	Cytoplasmic CsgF	$0.0$
$G_{c}$	Cytoplasmic CsgG	$0.0$
$A_{p}$	Periplasmic CsgA	$0.0$
$B_{p}$	Periplasmic CsgB	$0.0$
$C_{p}$	Periplasmic CsgC	$0.0$
$E_{p}$	Periplasmic CsgE	$0.0$
$F_{p}$	Periplasmic CsgF	$0.0$
$G_{p}$	Periplasmic CsgG	$0.0$
$E_{p9}$	Periplasmic CsgE (Nonamer)	$0.0$
$G_{p9}$	Periplasmic CsgG (Nonamer)	$0.0$
$GEF$	Complex of CsgGEF	$0.0$
$A_{e}$	Extracellular CsgA	$0.0$
$B_{e}$	Extracellular CsgB	$0.0$
$F_{e}$	Extracellular CsgF	$0.0$

Parameters

Table. 11.2.2.3. List of Parameters

Parameter	Value	Unit	Description	Reference
$k_{transRpoS}$	$3.74 \times 10^{-4}$	/sec	RpoS Transcription	^[24]
$k_{degRpoS}$	$3.85 \times 10^{-4}$	/sec	Natural degradation of RpoS	^[25]
$k_{degAc}$	$3.21 \times 10^{-5}$	/sec	Natural Degradation of cytoplasmic CsgA	^[26]
$k_{degBc}$	$3.21 \times 10^{-5}$	/sec	Natural Degradation of cytoplasmic CsgB	^[26]
$k_{degCc}$	$4.81 \times 10^{-5}$	/sec	Natural Degradation of cytoplasmic CsgC	Assumption
$k_{degDc}$	$6.42 \times 10^{-5}$	/sec	Natural Degradation of cytoplasmic CsgD	^[27]
$k_{degEc}$	$4.81 \times 10^{-5}$	/sec	Natural Degradation of cytoplasmic CsgE	Assumption
$k_{degFc}$	$4.81 \times 10^{-5}$	/sec	Natural Degradation of cytoplasmic CsgF	Assumption
$k_{degGc}$	$3.21 \times 10^{-5}$	/sec	Natural Degradation of cytoplasmic CsgG	^[26]
$k_{transAc}$	$9.21 \times 10^{-2}$	/sec	CsgATranscription	^[28]
$k_{transBGc}$	$2.14 \times 10^{-2}$	/sec	CsgB, C, D, E, F, G Transcription	^[28]
$k_{1A}$	$1 \times 10^{-2}$	/sec	Passage rate to Periplasm of CsgA	Assumption
$k_{1B}$	$1 \times 10^{-2}$	/sec	Passage rate to Periplasm of CsgB	Assumption
$k_{1C}$	$1 \times 10^{-1}$	/sec	Passage rate to Periplasm of CsgC	Assumption
$k_{1E}$	$1 \times 10^{-1}$	/sec	Passage rate to Periplasm of CsgE	Assumption
$k_{1F}$	$1 \times 10^{-1}$	/sec	Passage rate to Periplasm of CsgF	Assumption
$k_{1G}$	$1 \times 10^{-1}$	/sec	Passage rate to Periplasm of CsgG	Assumption
$k_{degAp}$	$3.21 \times 10^{-5}$	/sec	Natural Degradation of periplasmic CsgA	^[26]
$k_{degBp}$	$3.21 \times 10^{-5}$	/sec	Natural Degradation of periplasmic CsgB	^[26]
$k_{degCp}$	$4.81 \times 10^{-5}$	/sec	Natural Degradation of periplasmic CsgC	Assumption
$k_{degEp}$	$4.81 \times 10^{-5}$	/sec	Natural Degradation of periplasmic CsgE	Assumption
$k_{degFp}$	$4.81 \times 10^{-5}$	/sec	Natural Degradation of periplasmic CsgF	Assumption
$k_{degGp}$	$3.21 \times 10^{-5}$	/sec	Natural Degradation of periplasmic CsgG	Assumption
$K_i$	$1 \times 10^{-8}$	M	Inhibiting Constant of CsgC	Assumption
$k_{onE}$	$7.6 \times 10^{5}$	/(M $\times$ S)	Binding Constant of CsgE	^[25]
$k_{offE}$	$2.8 \times 10^{-3}$	/(M $\times$ S)	Releasing Constant of nonanoic CsgE	^[25]
$k_{onG}$	$7.6 \times 10^{5}$	/(M $\times$ S)	Binding Constant of CsgG	^[25]
$k_{offG}$	$2.8 \times 10^{-3}$	/(M $\times$ S)	Releasing Constant of nonanoic CsgG	^[25]
$k_{onGEF}$	$3.84 \times 10^{4}$	/(M $\times$ S)	Binding Constant of CsgGEF	^[22]
$k_{offGEF}$	$2.8 \times 10^{-3}$	/(M $\times$ S)	Releasing Constant of CsgGEF	^[22]
$k_{2A}$	$1 \times 10^{-2}$	/sec	Passage rate to extracellular of CsgA	Assumption
$k_{2B}$	$1 \times 10^{-2}$	/sec	Passage rate to extracellular of CsgB	Assumption
$k_{2F}$	$1 \times 10^{-1}$	/sec	Passage rate to extracellular of CsgF	Assumption
$k_{degAe}$	$8.37 \times 10^{-6}$	/sec	Natural Degradation of extracellular CsgA	^[29]
$k_{degBe}$	$8.37 \times 10^{-6}$	/sec	Natural Degradation of extracellular CsgB	^[29]
$k_{degFe}$	$1.20 \times 10^{-6}$	/sec	Natural Degradation of extracellular CsgF	Assumption
$k_{degEp9}$	$1.60 \times 10^{-6}$	/sec	Natural Degradation of nonanoic CsgE	Assumption
$k_{degGp9}$	$1.60 \times 10^{-6}$	/sec	Natural Degradation of nonanoic CsgC	Assumption
$k_{degGEF}$	$8.37 \times 10^{-6}$	/sec	Natural Degradation of CsgGEF	Assumption

Results

Simulations were performed to make quantitative analysis. Since many of the parameters were not existent, multiple assumptions have been made based on its biological role. The simulation results are shown here.

Fig11.2.2.2. Simulation Results for Part 2: Curli Display

The results reveal the time variation of protein expression in each process of curli display. This model provides a basic idea of the speed of Csg proteins proliferating to the extracellular.

On the other hand, due to the limited amount of parameters available, multiple assumptions were made to run this model. However, as these assumptions were made based on each component’s biological role, we believe the results do not contradict with the biological phenomenon. The results are applied to Part 3.

Part3. Curli Fiber Formation

Description of the Curli Fiber Formation

In this part, we modeled the process of Csg proteins forming Curli Fiber, using the expression levels shown in Part 2. The former part described the change in quantity of CsgA and B found in the extracellular, and these components are known to form Curli Fibers. CsgA is the main component for Curli Fiber formation, and CsgB connects CsgA to the membrane. Here, we quantitatively assessed the formation and dissociation of Curli Fiber.

Model Structure

Fig. 11.2.3.1. Overview: Curli Fiber Formation

The formation of Curli Fiber can be divided into four major steps, shown in Fig. 11.2.3.1.

Step 1: CsgA, B, F are released from the outer membrane to the extracellular.
Step 2: CsgB is fixed to the surface of the outer membrane by CsgF.
Step 3: CsgA forms amyloid on the outer membrane, polymerizes, then binds to CsgB.
Step 4. The amyloid satisfying the conditions form Curli Fibers.

Model Construction

1. Fixation of CsgB to the outer membrane by CsgF

CsgF and CsgB play crucial roles in curli fiber formation on the outer membrane^[1]. CsgF (in detail, the complex formed with nonamer of CsgE and CsgG,and dimer of CsgF) anchors CsgB to the membrane surface, establishing the foundation for curli fiber assembly. CsgB experiences multiple quantitative changes. One is the supply of CsgB from the periplasm, then anchored to the membrane, and the other is the return to free state from the membrane. Therefore, we define $B_{on}$ as the concentration of CsgB that is anchored to the outer membrane, and $B_{off}$ as the concentration of free CsgB in the extracellular environment.
The ODE of $B_{on}$ includes a term indicating the concentration change due to the formation and cleavage of the Curli Fiber.

\begin{align} \frac{d[B_{off}]}{dt}=&k_{2B}[B_{p}]-k_{degB_{e}}[B_{e}]-k_{bon}[B_{off}][GEF]+k_{boff}[B_{on}]\\ \frac{d[B_{on}]}{dt}=&k_{bon}[B_{off}][GEF]-k_{boff}[B_{on}]-k_{oncur}<pd>[A_{Ptot}][B_{on}]\notag\\ &+k_{offcur}[Cur]\\ \end{align}

2. Amyloid Synthesis

Since CsgA does not form amyloids at a small degree of polymerization but forms long amyloids at a larger degree, it is not realistic to create an ODE defining CsgA with different degrees of polymerization sequentially as variables. Therefore, we adopted a model in which the concentration of the amyloids’ bindable ends, amyloids with a degree of polymerization of two or more (or “polymers”), monomers, and dimers are used as variables^[30] ^[31].

Fig. 11.2.3.2. Figure of Amyloid Synthesis

In this model, the number of bindable amyloid ends decreases and the number of polymers increases over time. The concentration ratio of the polymer to the binding ends is used to illustrate the degree of polymerization, and the degree of polymerization increases as the reaction proceeds.
The number of amyloid ends bindable is altered by dimer formation and degradation, polymer-to-polymer end joining, and cleavage within the polymer. Specifically, when one dimer is formed from two monomers, two bindable ends will be formed. On the other hand, when two polymers join, the two bindable ends will be lost. When a polymer cleaves to form two new polymers, two new bindable ends are formed at the cleavage site.
As the reaction between polymers depends more on the probability of the binding ends colliding with each other than on the collision between polymers, the reaction is expressed as the square of the concentration of binding end $[A_{Etot}]$ .

\begin{align} \frac{d[A_{1}]}{dt}=&k_{2A}[A_{p}]-k_{degA_{e}}[A_{e}]-f_{N}[A_{1}]^2+b_{s1}[A_{2}]\\ &-f_{G}[A_{Etot}][A_{1}]+b_{s1}[A_{Ptot}]\notag\\ \frac{d[A_2]}{dt}=&f_{N}[A_{1}]^2-b_{s1}[A_{2}]-f_{J}[A_{Etot}]^2+b_{s2}[A_{Ptot}]\\ \frac{d[A_{Etot}]}{dt}=&2f_{N}[A_{1}]^2-2b_{s1}[A_{1}]-2f_{J}[A_{Etot}]^2+2b_{s2}[A_{Ptot}]\\ \frac{d[A_{Ptot}]}{dt}=&f_{N}[A_{1}]^2-b_{s1}[A_{2}]+f_{G}[A_{Etot}]^2-b_{s1}[A_{Etot}]\\ &-k_{oncur}<pd>[A_{Ptot}][B_{on}]+k_{offcur}[Cur]\notag\\ <pd>=&\frac{[A_{Ptot}]}{[A_{Etot}]}\\ \end{align}

3. Model of Curli Fiber Formation

The formation of Curli Fiber is exclusive to amyloids that are decently long. Thus, by using the degree of polymerization, we illustrated that Curli Fibers will not be formed if they are not polymerized, even if the polymer concentration $[A_{ptot}]$ increases.

\begin{align} \frac{d[Cur]}{dt}&=k_{oncur}<pd>[A_{Ptot}][B_{on}]-k_{offcur}[Cur]\\ \end{align}

Abbreviations

Table. 11.2.3.1. List of Abbreviations

Abbreviation	Description
$B_{on}$	Bonded state of CsgB
$B_{off}$	Dissociated state of CsgB
$k_{b_{on}}$	Binding Constant of CsgB
$k_{b_{off}}$	Dissociation Constant of CsgB
$k_{oncur}$	Binding Constant of Curli
$k_{offcur}$	Dissociation Constant of Curli
$<pd>$	Polymer Degree
$A_{Ptot}$	Total polymer concentration of CsgA
$A_{Etot}$	Total bindable end concentration of CsgA
$A_{1}$	Monomer CsgA concentration
$A_{2}$	Dimer CsgA concentration
$f_{N}$	Binding rate of monomer
$f_{G}$	Binding rate of polymer + monomer
$f_{J}$	Binding rate of polymer + polymer
$b_{s1}$	Dissociation rate to monomer
$b_{s2}$	Dissociation rate to two polymers
$Cur$	Concentration of Curli

Initial Conditions

Table. 11.2.3.2. List of Initial Conditions

Abbreviation	Description	Initial Value
$B_{off}$	Dissociated state of CsgB	$0.0$
$B_{on}$	Bonded state of CsgB	$0.0$
$A_{1}$	Monomer CsgA concentration	$0.0$
$A_{2}$	Dimer CsgA concentration	$0.001$
$A_{Ptot}$	Total polymer concentration of CsgA	$0.0$
$A_{Etot}$	Total bindable end concentration of CsgA	$0.001$
$Cur$	Concentration of Curli	$0.0$

Parameters

Table. 11.2.3.3. List of Parameters

Parameter	Value	Unit	Description	Reference
$k_{b_{on}}$	$0.00425$	$s^{-1}$	Binding Constant of CsgB	^[5]
$k_{b_{off}}$	$1.0 \times 10^{-4}$	$s^{-1}$	Dissociation Constant of CsgB	Assumpiton
$k_{oncur}$	$0.045$	$(mol \times s)^{-1}$	Binding Constant of Curli	^[5]
$k_{offcur}$	$1.1 \times 10^{-4}$	$s^{-1}$	Dissociation Constant of Curli	^[5]
$f_{N}$	$24000$	$(mol \times s)^{-1}$	Binding rate of monomer	^[6]
$f_{G}$	$24000$	$(mol \times s)^{-1}$	Binding rate of polymer + monomer	^[6]
$f_{J}$	$1.317$	$(mol \times s)^{-1}$	Binding rate of polymer + polymer	^[6]
$b_{s1}$	$2.56 \times 10^{-4}$	$s^{-1}$	Dissociation rate to monomer	^[7]
$b_{s2}$	$2.56\times 10^{-4}$	$s^{-1}$	Dissociation rate to two polymers	^[7] Assuming equal to $b_{s1}$

Results

Fig. 11.2.3.3. Result of curli fiber formation

The results include the variables from Part 1,2, such as the expression of BIND-PETase from plasmid and extracellular CsgA and B. It shows that the formation of curli fibers gradually starts after the CsgA and B have been transported to the extracellular. This is due to the scale of rate constants, such as dimerization and eloganation reaction of CsgA, which are significantly larger than those with other rate constants involved in Curli Fiber formation. As a result, the concentration of Curli Fibers became evidently small compared to its main component, CsgA. This indicates that the process of Curli fiber formation is constrained by a low binding probability.

To solve this problem, we believe the expression of CsgEFG is necessary. As the rate-limiting step of Curli Fiber formation is the fixation of CsgB on the outer membrane, the increase in expression of CsgEFG may enhance the formation of it. In total, the results have demonstrated the whole scheme of gene expression to Curli Fiber formation in theory, providing insights for lab experiment application.

Part4. PET Degradation

Description of the PET Degradation Model

In this part, we modeled the process of PET degradation by BIND-PETase on the Curli Fiber formed in Part 3. In the lab experiment, we were able to identify PET degradation by BIND-PETase. Therefore, in the lab, we examined the time variation of PET degradation for BIND-PETase to degrade PET in social implementation. In this project, PETase is anchored to the Curli Fiber, which is difficult to reproduce with models such as those produced by conventional enzyme reactions (e.g. Hill equation) ^[32]. Thus, we constructed a new model tailor-made for our project, which can be applied to biological phenomena whose Hill equation is not applicable.

Model Structure

1. PET-PETase binding, and cleavage probability model

In this section, we applied our knowledge of thermodynamics and physical chemistry to derive the binding probability $k(t)$ and cleavage probability $\gamma (t)$ of PET and PETase, which are necessary for the PET degradation model. Strictly speaking, $k(t)$ and $\gamma (t)$ depend on many variables. However, it is difficult to estimate which parameter contributes due to a vague understanding of the reaction mechanism. That is why we write $k(t)$ and $\gamma (t)$ look like variables but calculated as constant, as follow.

Using the Maxwell-Boltzmann distribution, the fraction of molecules with velocity $v$ is defined as the fraction of molecules with velocity $f(v)$ . The fraction of molecules $f(v)$ with velocity $v$ , is calculated as follows.

\begin{align} f(v)&=4\pi\left(\frac{m}{2\pi{k_bT}}\right)^{\frac{3}{2}}v^2\exp{\left(-\frac{mv^2}{2\pi{k_bT}}\right)}\\ \int_{0}^{\infty}f(v)dv&=1\\ \end{align}

We now consider the decomposition reaction of PET by PETase. When the mass of PETase is equal to $m$ and the kinetic energy is equal to $E_a$ , the velocity $v$ of thermal motion of PETase molecules is calculated as follows.

\begin{align} \frac{mv^2}{2}&=\frac{E_a}{N_a}\\ v&=\sqrt{\frac{2E_a}{mN_a}} \end{align}

Therefore, the fraction of molecules with kinetic energy above the activation energy $E_a$ can be calculated as follows.

\begin{align} \int_{\sqrt{\frac{2E_a}{mN_a}}}^{\infty}f(v)dv\ &=\int_{\sqrt{\frac{2E_a}{mN_a}}}^{\infty} \left\{4\pi\left(\frac{m}{2\pi{k_bT}}\right)^{\frac{3}{2}}v^2\exp{\left(-\frac{mv^2}{2\pi{k_bT}}\right)}\right\} dv\\ \end{align}

Here, the binding probability $k(t)$ and the cleavage probability $\gamma (t)$ are calculated as follows, by substituting the activation free energies required for the binding and cleavage of PET by PETase, respectively.

\begin{align} k(t)&=\int_{\sqrt{\frac{2\times 20.0\times10^3\times 4.184}{mN_a}}}^{\infty}f(v)dv=1.7572579752\times10^{-14}\\ γ(t)&=\int_{\sqrt{\frac{2\times 15.1\times10^3\times 4.184}{mN_a}}}^{\infty}f(v)dv=5.7167033890\times10^{-11} \end{align}

2. Length of PET to be cut off

Fig. 11.2.4.1. Defining Quantity for modeling PET Degradation

As shown in Fig. 11.2.4.1, there are two conditions of PET decomposition by PETase on the Curli Fiber. The first case is when Curli Fiber binds to PET at the expected width value $E_d$ , then no decomposition by PETase occurs outside the expected width $E_d$ . The other case is when PETase decomposes PET inside the width $E_d$ . We made the following assumptions to consider the two cases.

!

Assumption 4.1: The expected width value $E_d$ of PET becoming degraded is shown below.

Detailed explanation of Assumption 4.1

The expected width value $E_d$ of degraded PET is shown below.

\begin{align} E_{d} = \sum_{n=1}^{\left\lfloor \frac{L}{l} \right\rfloor} \left\{ k(t) \gamma(t) \right\}^2 \left\{ 1 - k(t) \gamma(t) \right\}^{\left\lfloor \frac{L}{l} \right\rfloor - n} \times (n - 1) l\\ \end{align}

Fig. 11.2.4.2. Image of cutting out PET

The calculation of $E_d$ can be done considering the following three scenarios.

Red (number {1,n})

Chemically bonded to PET
Cut PET

Therefore, the probability is,

\begin{align} k(t)\gamma(t)\space(n=1)\\ \lbrace{k(t)}\gamma(t)\rbrace^2(n\geq2) \end{align}

Green (number {2,3, …, n-1}

Chemical bond with PET is inconsequential
PET cut is inconsequential

Therefore, the probability is not necessary to be considered.

Orange

Not chemically bonded to PET

Therefore, the probability is,

\begin{align} \lbrace1-k(t)\gamma(t)\rbrace^{\left[\frac{L}{l}\right]-n} \end{align}

Next, we assumed that the total length per Curli Fiber is $L$ and that there are PETases on it with an interval of $l$ . The number of PETases on a single Curli Fiber can then be expressed as $\left\lfloor\frac{L}{l}\right\rfloor$ using the floor function. The width of the PET to be cut out at $n = 1, 2, \dots, \left\lfloor\frac{L}{l}\right\rfloor$ assigned to the PETase is $0, l ... , (n-1)l, ... ,(\left\lfloor\frac{L}{l}\right\rfloor-1)l$ . Therefore, $E_d$ is calculated as follows.

\begin{align} E_d&=k(t)\gamma(t)\lbrace1-k(t)\gamma(t)\rbrace^{\lfloor\frac{L}{l}\rfloor-1}\times0\notag\\ &+\lbrace{k(t)\gamma(t)}\rbrace^2\lbrace1-k(t)\gamma(t)\rbrace^{\lfloor[\frac{L}{l}\rfloor]-2}\times{l}\notag\\ &+\lbrace{k(t)\gamma(t)}\rbrace^2\lbrace1-k(t)\gamma(t)\rbrace^{\lfloor\frac{L}{l}\rfloor-3}\times{2l}\notag\\ &+\dots\notag\\ &+\lbrace{k(t)\gamma(t)}\rbrace^2\lbrace1-k(t)\gamma(t)\rbrace^{\lfloor\frac{L}{l}\rfloor-n}\times{(n-1)l}\notag\\ &+\dots\notag\\ &+\lbrace{k(t)\gamma(t)}\rbrace^2\lbrace1-k(t)\gamma(t)\rbrace^{0}\times\left\{\left(\lfloor\frac{L}{l}\rfloor-1\right){l}\right\}\\ &=\sum^{\lfloor\frac{L}{l}\rfloor}_{n=1}\lbrace{k(t)\gamma(t)}\rbrace^2\lbrace1-k(t)\gamma(t)\rbrace^{\lfloor\frac{L}{l}\rfloor-n}\times{(n-1)l}\\ \end{align}

!

Assumption 4.2: The time $t_j$ at which the $j$ th PET cleavage reaction occurs is expressed as below.

\begin{align} t_0=0\space,\space{t_R=t_i}\\ t_j=j\Delta{t}\space{(j=0,1,\dots,i)} \end{align}

Detailed explanation of Assumption 4.2

****Fig. 11.2.4.3.**** Division of time scale $[0, t_R]$

When PET is degraded, the degree of polymerization of PET gradually decreases and ultimately converges to 1. $t_R$ is set as the time when the degree of polymerization reaches 1, to consider dividing the time interval [0, $t_R$ ] into $i$ segments for each $\Delta t$ of the cleavage reaction time (Fig. 11.2.4.3). If $\Delta t=\frac{t_R}{i}$ , the $j$ th cleavage reaction can occur at time $t_j$ as follows.

\begin{align} t_0=0\space,\space{t_R=t_i}\\ t_j=j\Delta{t}\space{(j=0,1,\dots,i)} \end{align}

We first considered the case where no degradation by PETase occurs outside the first width $E_d$ . By setting the average of the lengths of PET molecules whose chain length at time $t$ is longer than the expected value $E_d$ , as $R(t)$ , it could be formulated as below.

\begin{align} R(t)\geq{E_d}\space\space{(\forall{t}\in[0,t_j])} \end{align}

Next, we will consider the case where degradation by PETase occurs within the width $E_d$ . We define $r_j(t)$ as the average length of PET molecules cut from $R(t_j)$ at time $t_j$ , where the chain length of the PET molecules at $t_j+t$ is shorter than the expected value $E_d$ . The formula is shown below.

\begin{align} r_j(t)\leq{E_d}\space{(\forall{t}\in[0,t_j],\space(j=0,1,\dots,i))} \end{align}

3. PET degradation model in length $R(t)$

Fig. 11.2.4.4. Degradation model of PET at the length of R(t)

!

Assumption 4.3: The expected value of the number of Curli Fibers $m(t)$ joining a PET of length $R(t)$ is as below.

Detailed explanation of Assumption 4.3

The expected value of the number of Curli Fibers $m(t)$ joining a PET of length $R(t)$ is as below.

\begin{align} m(t) =\frac{P(t)\left\{{1-P(t)^{\left[\frac{R(t)}{d(t)}\right]}}\right\}}{(1-P(t))^2} - \frac{\left[\frac{R(t)}{d(t)}\right]{P(t)}^{\left[\frac{R(t)}{d(t)}\right]+1}}{1-P(t)} \end{align}

As shown in Fig. 11.2.4.4, multiple Curli Fibers join PET of the length $R(t)$ in the width $E_d$ . The number of Curli Fibers that can be coupled is $1, 2, \dots, \left\lfloor\frac{R(t)}{d}\right\rfloor$ , using the floor function. Therefore, the expected value of the number of Curli Fibers $m(t)$ that bind to a PET of length $R(t)$ can be calculated as follows.

\begin{align} m(t)&=\lbrace{k(t)\gamma(t)}\rbrace^2\times1\notag\\ &+\lbrace{\lbrace{k(t)\gamma(t)}\rbrace^2}\rbrace^2\times2\notag\\ &+\dots\notag\\ &+\lbrace{\lbrace{k(t)\gamma(t)}\rbrace^2}\rbrace^{n}\times{n}\notag\\ &+\dots\notag\\ &+\lbrace{\lbrace{k(t)\gamma(t)}\rbrace^2}\rbrace^{\left\lfloor {\frac{R(t)}{d(t)}} \right\rfloor}\times\left\lfloor \frac{R(t)}{d(t)} \right\rfloor\notag\\ &=\sum^{\left\lfloor \frac{R(t)}{d(t)} \right\rfloor}_{n=1}n\times{P(t)}^n\\ &\left(P(t)\stackrel{\mathrm{def}}{=}\lbrace{k(t)\gamma(t)}\rbrace^2\right) \end{align}

For PET with a length of $R(t)$ , Curli Fibers of width $E_d$ at a number of $m(t)$ .
Since the time for cutting of PET takes $\Delta t$ , $R(t+\Delta t)$ is the length after cutting and $m(t)+1$ is the total number of PET molecules formed. Therefore, the following formulas in length can be established.

\begin{align} R(t)=m(t)\times{d(t)}+\lbrace{m(t)+1}\rbrace\times{R(t+\Delta{t})}\\ \end{align}

For the degree of polymerization $a_R(t)$ in PET of length $R(t)$ , this relation is applicable.

\begin{align} \frac{R(t)}{R(0)}=\frac{a_R(t)}{a_R(0)}\\ a_R(t)=\frac{a_R(0)}{R(0)}R(t) \end{align}

For the concentration $c_R(t)$ in PET of length $R(t)$ , this relation is applicable.

\begin{align} {R(0)}{b_R(0)}={R(t)}{b_R(t)}\\ b_R(t)=\frac{R(0)b_R(0)}{R(t)} \end{align}

4. PET degradation model in length $r_j(t)$

Fig. 11.2.4.5. Degradation model of PET at the length of $r_j(t)$

First we will consider the degradation of $r_0(t)$ . As shown in Fig. 11.2.4.5, if the length of each Curli Fiber is $L$ and the spacing between PETases aligned on the Curli Fiber is $l$ , the number of PETases that bind and break per PET molecule of length $r_0(t)$ at a given time $t$ is as below.

\begin{align} &\frac{k(t)\gamma(t)r_0(t)}{l} \end{align}

Thus, when considering a PET molecule of length $r_0(t)$ at the time $t$ , $\Delta t$ later, will be separated into a quantity as shown.

\begin{align} &\frac{k(t)\gamma(t)r_0(t)}{l}+1 \end{align}

In total, the degradation of $r_0(t+\Delta t)$ can be explained as follows.

\begin{align} r_0(t+\Delta{t})=\frac{r_0(t)}{\frac{k(t)\gamma(t)r_0(t)}{l}+1}\\ \Delta{r_0(t)}\stackrel{\mathrm{def}}{=}r_0(t+\Delta{t})-r_0(t) \\ \Delta{r_0(t)}=\frac{-k(t)\gamma(t)r_0(t)^2}{k(t)\gamma(t)r_0(t)+l} \end{align}

For the degree of polymerization $a_{r_0}(t)$ in PET of length $r_0(t)$ , this relation is applicable.

\begin{align} \frac{r_0(t)}{r_0(0)}&=\frac{a_{r_0}(t)}{a_{r_0}(0)}\\ a_{r_0}(t)&=\frac{a_{r_0}(0)}{r_0(0)}r_0(t)\\ \end{align}

On top of that, for the concentration $c_{r_0}(t)$ of PET of length $r(t)$ in solution, this relation is applicable.

\begin{align} {r_0(0)}{c_{r_0}(0)}={r_0(t)}{c_{r_0}(t)}\\ c_{r_0}(t)=\frac{{r_0(0)}{c_{r_0}(0)}}{r_0(t)} \end{align}

From here, we will consider a more general degradation of $r_j(t)$ .

By Assumption 4.4, the degradation of length $r_j(t+\Delta t)$ can be illustrated as follows. Though $k(t)$ and $\gamma (t)$ are constant in this model, technically it should be variable, and thus it is written down as a function with variable $t$ .

\begin{align} r_j(t+\Delta{t})=\frac{r_j(t)}{\frac{k(t+t_j)\gamma(t+t_j)r_j(t)}{l}+1} \end{align}

For the degree of polymerization $a_{r_j}(t)$ in PET of length $r_j(t)$ , this relation is applicable.

\begin{align} \frac{r_j(t)}{r_j(0)}=\frac{a_{r_j}(t)}{a_{r_j}(0)}\\ a_{r_j}(t)=\frac{a_{r_j}(0)}{r_j(0)}r_j(t)\\ \end{align}

Similarly, for the concentration $c_{r_j}(t)$ of PET of length $R(t)$ in solution, the relation is as follows.

\begin{align} {r_j(0)}{c_{r_j}(0)}={r_j(t)}{c_{r_j}(t)}\\ c_{r_j}(t)=\frac{{r_j(0)}{c_{r_j}(0)}}{r_j(t)}\\ \end{align}

Lastly, considering the conditions of $j=i$ may become crucial to know the termination of the PET degradation.

\begin{align} a_{r_i}(t)\geq1 \space(t\leq{t_R})\\ a_{r_i}(t)=1 \space(t\geq{t_R}) \end{align}

5. Total PET Concentration change by PETase

The total concentration of PET $c(t)$ can be formulated as below, at the state of $t\in [t_j, t_j+1] (j=1, 2, . . . , i)$ .

\begin{align} c(t)=b(t)+\sum^{j}_{j'}\beta_{j'}(t-j'\Delta{t})\\ \end{align}

6. The situation in Width $E_d \leq l$

When $E_d \leq l$ , it means that only one or less PETase out of one Curli Fiber can bind to PET. Therefore, the case distinction between $R(t)$ and $r_j(t)$ cannot occur when $E_d<l$ . Since there is no case distinction, the length of PET at this time is expressed only by $r(t)$ , and we would like to see the time variation of this length.

The number of PET sites that can be cut by PETase can be understood by using the degree of polymerization $a(t)$ , which is $a(t)-1$ . Therefore, PET is divided into $k(t)\gamma(t)(a(t)-1)$ after one cleavage reaction.

\begin{align} r(t+\Delta{t})=\frac{r(t)}{k(t)\gamma(t)(a(t)-1)+1} \end{align}

Abbreviations

Table. 11.2.4.1. List of Abbreviations

Abbreviations	unit	Description
$E_d$	m	the expected value of the length of PET cut out
$R(t)$	m	Length of a PET molecule at time $t$ , whose chain length is longer than $E_d$ ( $E_d \geq l$ )
$r_j(t)$	m	Length of PET molecule at time $t_j+t$ , which PET cleaved from a PET molecule of chain length $R(t)$ at time $t_j$ , and shorter than $E_d$ ( $E_d \geq l$ )
$r(t)$	m	Length of a PET molecule at time $t$ ( $E_d \leq l$ )
$a_R(t)$		Average degree of polymerization of PET with the length of $R(t)$
$a_{r_j(t)}$		Average degree of polymerization of PET with the length of $r_j(t)$
$c_R(t)$	mol/L	Concentration of PET with the length of $R(t)$
$c_{r_j}(t)$	mol/L	Concentration of PET with the length of $R(t)$
$c(t)$	mol/L	Total concentration of PET

Parameters

Table. 11.2.4.2. List of Parameters

Constant	value	unit	Description
$m$	$29\times1.66033 \times10^{−24}$ ^[33]^[34]	kg	Mass of PETase
$k_b$	$1.380649\times10^{23}$ ^[35]	m^2 kg s^(-2) K^(-1)	Boltzmann constant
$T$	$300$	K	Temperature
$E_a$	PET-PETase binding: $\\$ $20.0\times4.184\times10^3$ ^[36] $\\$ PET cleavage: $\\$ $15.1\times4.184\times10^3$ ^[36]	J/mol	Activation free energy
$N_a$	$6.02\times10^{23}$ ^[37]	mol^(-1)	Avogadro constant
$k(t)$	$1.7572579752\times10^{-14}$		(Calculated by Maxwell–Boltzmann distribution)
$γ(t)$	$5.716693785\times10^{-11}$		(Calculated by Maxwell–Boltzmann distribution)
$l$	$19$ ^[38]	Å
$[\frac{L}{l}]$	$8$ ^[38]

Results

Fig. 11.2.4.7. Simulation of $R(t)$ and $r_j(t)$

Firstly, we demonstrated the fundamental behavior of the model by applying large values for $k(t)$ and $\gamma(t)$ , by assigning the orders around $10^{-3}$ to $10^{-5}$ . In Fig. 11.2.4.7, the PET length decreases and converges to zero, which aligns with the assumption of the mathematical model. Therefore, this new constructed model for PET degradation accurately represents the degradation of PET by PETase on Curli Fiber, confirming the model’s quantitative validity.
Having proven that the formulas of our model are likely correct, we then input $k(t)$ and $\gamma (t)$ with the values we estimated. We tried to analyze the result of our modeling, however, due to low magnitudes of both parameters, we were unable to run the computational simulation using Python. The simulation failed to generate a time scale, likely due to limits of our computers’ capabilities. Consequently, a higher-performance computer is required to properly output the simulation results. Despite this, this model successfully demonstrates that our project can execute the PET degradation process.

In Part 4, we described a new model to explain the PET degradation system for BIND-PETase. This proved that PET degradation is done by BIND-PETase, which resulted in the same conclusion with the results from the lab experiment. Since this model does not use Hill equations, it can illustrate reactions where enzymes are not freely diffused, that it can be applied to other whole-cell catalysts.

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