Model - iGEM 2020 Team:Waseda

Modeling

Zombie vs Samurai circuit

Our team is eligible for this prize because we actively used dry-wet cycles.  We designed a genetic circuit that realizes a bistable system switched by the number of cells.  Before implementing the genetic circuit in a wet experiment, we modeled it and simulated the behavior.  Through the simulation, we discovered some problems like degradation time and crosstalk ratio, so we introduced new technology and verified its usefulness by conducting wet experiments.  Based on the results, we re-simulated and confirmed that the designed system works. Moreover, we devised a way to explain this modeling ingeniously. Modeling plays an essential role in synthetic biology, but it is difficult to explain it accurately.  To solve this problem, we likened this bi-stable system to the war between zombies and humans.  We are convinced that this interesting story attract anyone, including the general public, and make the system easier to understand.

Our Zombie vs Samurai Circuit is as shown in Fig.4-1-1. To demonstrate this model, we solved the ordinary differential equations containing Hill repression and activation terms. The ideal results for the war between Zombies and Samurai are in the Fig.4-1-2. This ideal result has previously been achieved with ordinary toggle switch genetic circuits without cell-cell communication, so we tried to develop this simulation and realize a toggle switch including cell-cell communication. Since the nullcline indicates the concentration when it does not change with time, the number of shared points of two nullclines corresponds to the number of stable states. As shown in Fig.4-1-2, there are three shared points on the nullcline and two of them are stable equilibrium points, so the system is bistable.

Fig.4-1-2 the ideal results of our gene circuit

4.2 The basic mothod and results of our genetical circuit

Before modeling of in vitro systems with cell-cell communication, we established differential-equations (1) to (6) (Fig.4-1-3) based on modeling for the same circuit which works in living E. Coli cells because of accumulated cases for such in vivo models [Ryoji S. et al., 2011 and iGEM tokyo_tech, 2013]. We formulated equations assuming that the system has a sufficient amount of R protein and I proteins activate the corresponding promoters directly (Fig.4-1-4). The meaning of each parameter is as shown in Table.4-1-1.

Fig.4-1-3 assumed Circuit; I protein directly activates promoter — Fig.4-1-3 Basic equations of our gene circuit

Table.4-1-1 description of each parameter

parameter	description
α1	Maximum expression of rate LacI, when PLux is activated by C6
α2	Maximum expression of rate LacI, when Pλ is repressed by CI
α3	Maximum expression of rate CI, when PRhl is activated by C4
α4	Maximum expression of rate LacI, when PLac is repressed by LacI
α5	Maximum expression of rate LuxI, when Pλ is repressed by CI
α6	Maximum expression of rate RhlI, when PLac is repressed by LacI
K1	z concentration at which the expression rate of LacI is half ofα1
K2	CI concentration at which the expression rate of LacI is half ofα2
K3	w concentration at which the expression rate of CI is half ofα3
K4	LacI concentration at which the expression rate of CI is half ofα4
K5	CI concentration at which the expression rate of LuxI is half ofα5
K6	LacI concentration at which the expression rate of RhlI is half ofα6
Nk (k=1,2,3,4,5,6)	Hill coefficient
Dk (k= LacI, CI LuxI RhlI, z, w)	Degradation rate
λz,λw	ratio of AHL produced by cell i depending on the concentration of I protein

Table.4-1-2 basic parameter set

parameter	value	unit
α1	100	ｎM/min
α2	500	ｎM/min
α3	100	ｎM/min
α4	100	ｎM/min
α5	400	ｎM/min
α6	100	ｎM/min
K1	27	ｎM
K2	30	ｎM
K3	6	ｎM
K4	50	ｎM
K5	30	ｎM
K6	50	ｎM
nk (k=1,2,3,4,5,6)	2	-
dk (k= LacI, CI LuxI RhlI, z, w)	1	/min
λz,λw	1	-

Fig.4-1-4 the basic nullcline of LacI and CI

At the same time, we drew Fig.4-1-5 as the time-course of each molecular concentration by using the same parameter set. It also showed there were two final stable states when the initial concentration of AHL was changed. Other initial values of this simulation were as shown in Table.4-1-3.

Fig.4-1-6 illustrates the behavior of our gene circuit. When the zombie state cell is induced with a samurai signal, the cell fate will change depending on the size of the signal. If the zombie receives a large number of samurai signals, the zombie will not be able to withstand the signal and will turn into a samurai. However, if samurai signal received by the zombie state cell is not enough, the zombie can revert back to being a zombie again, without completely transforming into a samurai. As a result, we were able to create two final states simply by changing the initial concentration of communication molecules.

Fig.4-1-5 the time-course of each molecule concentration

Fig.4-1-6 the concept behavior of our gene circuit

Table.4-1-3 basic initial values

variable	value	unit
LacI	80	ｎM
CI	70	ｎM
LuxI	80	ｎM
RhlI	80	ｎM

In contrast to cellular system, a cell-free system lacks cell growth and has much smaller degradation- dilution term, which lead to longer operating time. To solve this problem, we first investigated the size of the required degradation term. Operation time for the state switch was estimated from the time that high concentration of Samurai repressors at one equilibrium point are reduced by degradation to the repressor concentration at another equilibrium point. Although we did not consider the term of synthesis in this calculation, there was no problem because this assumption without protein synthesis shows shorter operation time than a case with protein synthesis. By using parameters from reference (Maurizi MR,1992) , we thus estimated operating time for state-switch between Zombie and Samurai in our cell-free system. We used the value of the equilibrium point of CI in Fig.4-1-5 and hold the equations (13) to (15). The time t1 at which the system reached one equilibrium point and the time t2 at which the system reached the other equilibrium point are expressed by equations (13) and (14), respectively. Besides, the difference between t1 and t2 is expressed as t by equation (15). We illustrated Fig.4-1-7 as the relationship between this degradation and time (equation (15)) and found that it takes too much time (approximately 20x104 min = 300 hours) to operate a state-switch between Samurai and Zombie. The blue point is about normal protein degradation, and the green point shows the case of the LVA degradation tag are introduced on the protein. This estimation of switching time implies the importance of repressor production in our system.

Fig.4-1-7 degradation ratio and time of transition between equilibrium points

Our simulation can show the fate of the battle between Zombie and Samurai in the test tube(Fig.4-1-8). Depending on initial concentration of components, we can prepare a Zombie test tube and a Samurai test tube which are stable state. Then, by mixing Zombie and Samurai cell-free solutions in various ratios, the battle started. To realize this battle, we used the same parameter as Table.4-1-2 and Table.4-1-3, and created datasets which were stabilized in two states with different values. We took the final point of dataset, added it together in various ratios such as 8:2 or 7:3, and then had it calculated again as a new initial value. As a result, we succeeded in showing that the mixing ratios produced two states. When Samurai state cell-free solution and Zombie state cell-free solution were mixed at a ratio of 8:2, the mixed solution was stable in the Samurai state. This means that the Samurai have beaten the Zombie. However, a slight change of the ratio completely turned their fate. When the Samurai and Zombie solutions were mixed at a ratio of 7:3, the mixed solution was stable in the Zombie state and the Zombie will destroyed the Samurai.

Fig. 4-1-8 basic war in the cell-free system

For artificial genetic circuits intercellular communication, we had to consider two types of signal crosstalks; one is the activation of the R protein with different communication molecules and the other is the activation of the promoter with different R proteins. Especially, we dealt with and modeled R　protein and Promoter crosstalk.

We needed to set the R protein concentration to see the crosstalk between AHL and R protein, so the Fig.4-1-3 was modified to Fig.4-1-9, and we consider the new reaction shown in Fig.4-1-10.

Fig.4-1-9　Improved circuit for crosstalk

We hold formulated equations (16) to (21), which is similar to equations (1) to (6), but we changed the first terms of equations (16) and (17) compared to equations (1) and (2). Thinking of R protein binding to DNA, we were able to regard R proteins as competitive inhibition each other. We assumed there was no great difference in promoter activation between any activated proteins, so we simply summed them up, forming equations (16) to (17). Using the parameters shown as Table.4-1-4, we drew a nullcline as Fig.4-1-11 by setting the change with time 0 of equations (16) to (21).

Fig.4-1-10　Improved circuit diagram for AHL-R protein crosstalk

Table.4-1-4 parameter set to AHL-R protein crosstalk

parameter	value	unit
α1	500	ｎM/min
α2	80	ｎM/min
α3	90	ｎM/min
α4	400	ｎM/min
α5	100	ｎM/min
α6	100	ｎM/min
α7	100	ｎM/min
α8	100	ｎM/min
k1	50	ｎM
k2	6	ｎM
k3	20	ｎM
k4	50	ｎM
k5	30	ｎM
k6	50	ｎM
k7	6	ｎM
k8	27	ｎM
nk(k=1,2,3,4,5,6)	2	-
dk(k= LacI, CI LuxI RhlI, z, w)	1	/min
λz,λw	1	-

The actual measurements of the wet experiments showed that there was severe crosstalks between R proteins and promoters. Even with the identified severe crosstalks between Rhl promoter and LuxR-3OC6HSL complex, reduction of maximum activity of Rhl promoter allowed us to prepare test tubes for Samurai and Zombie. Because of the same activation by samurai signaling molecule for the both repressors, Samurai state seemed not to be stable (Fig.4-1-11 center). However, we could reduce CI repressor production from Rhl promoter by mutations at its DNA sequence. Even though such reduction had a risk of unitability of Zombie state, adequate modulation of maximum activity of Rhl promoter kept the stability. When we reduce the maximum expression by Rhl promoter to 1/10 compared with Lux promoter activity, phase space analysis show both stabilities of Zombie and Samurai state; the reduction did not change the number of intersections of nullclines (Fig.4-1-11 right). Moreover, for this model we drew Fig.4-1-12 as a time course of mixed Zombie and Samurai. We found that even if we consider crosstalk, the two powers can compete and be biased to either side depending on their ratios.

Fig.4-1-11 phase space analysis of the severe crostalk between R proteins and Promotes

Fig.4-1-12 war in the cell-free system containing crosstalk between R proteins and

Considering these modeling results, Wet-Dry cycle showed that our cell-free system works as intended in the scenario.

Reference

[1] Maurizi MR. Proteases and protein degradation in Escherichia coli. Experientia. 1992 Feb 15 48(2):178-201

Multi IFFL

In the incoherent feed forward loop(Fig 4-2-1), the expressions of each gene: x,y, and z are formulated as the differential-equation (1), (2) and (3).

Table 4-2-1 description of each parameter of IFFL

Variables	Description	unit
[x]	Concentration of protein x	µM
[y]	Concentration of protein y	µM
[z]	Concentration of protein z	µM

Parameters	Description	unit
a1	Maximum expression rate of protein X	µM/min
a2	Mamimum expression rate of protein Y, when promoter of z is activated by protein X	µM/min
αx	Mamimum expression rate of protein Z, when promoter of z is activated by protein X	µM/min
αy	Mamimum expression rate of z, when promoter of protein Z is activated by protein Y	µM/min
K1	x concentration at which the expression rate of y is half of α2	µM
Kx	x concentration at which the expression rate of y is half of αx	µM
Ky	x concentration at which the expression rate of y is half of αy	µM
βk(k=1,x,y)	Hill coefficient	-
dk(k=x,y,z)	Degradation rate	/min

We designed an expanded IFFL system containing two target genes (Z1,Z2) each of which with different peak time to the other. We call this Multi -target- IFFL, or Multi IFFL. By adjusting the parameters for each activation and repression, we can provide a time difference in the generation of each pulse. To check whether the designed multi-target IFFL can really generate two pulses with a time delay, we extensively searched for parameter space for the system.

We first investigated how the concentration of Z changes over time when the parameters Kx and Ky are changed independently in Formula (3). When searching, we varied Kx and Ky from 1 to 1000 every 100 and other parameters were fixed as shown on Table-4-2-2, Table-4-2-3 respectively.

Fig 4-2-3 Kx and Ky search independently

Table 4-2-2 Fixed parameters in the Kx search

parameter	value
ax	4000
βx	1
βy	2
Ky	50

Table 4-2-3 Fixed parameters in the Ky search

parameter	value
ax	4000
βx	1
βy	2
Kx	8

Next, we constructed a heatmap to check how much time (t) of the pulse peak is delayed as we simultaneously changed the parameters Kx and Ky in the formula (3). Fig. 4-2-4 is the results. Other parameters are fixed as shown on Table 4-2-4.

Table.4-1-4 parameter set to AHL-R protein crosstalk

parameter	value
ax	4000
βx	1
βy	2

Furthermore, in order to evaluate the balance of the pulse shape, we calculated the ratio of the peak concentration to the following steady-state concentration for each parameter and presented it in a heat map. This is called the ratio heatmap. (Fig 4-2-5)

Based on these two indicators, we multiplied the cross heatmap by the value of the time heatmap and the ratio heatmap for each parameter to find out which parameter has both time difference and balance, comprehensively .To make this cross heatmap, we subtracted a certain constant number of time values from the time heatmap because we have to make the time delay above a certain level. If the time heatmap is negative, it is treated as 0. This is called time processed heatmap. Fig 4-2-6 is the time processed heat map, calculated by subtracting 4 from the all values of the time heatmap.

The values of the cross heatmap were calculated as the formula (5)

We selected the optimal parameter for the multi IFFL in the resulting match heatmap as the one with the highest score. Two pulses were generated using the optimal parameter in practice (Fig 4-2-7)(Table 4-2-5, 4-2-6). Yellow indicates the temporal variation of the concentration of Z1 and green indicates that of Z2. As a result, we succeeded in creating pulses with a time difference.

Table 4-2-5

parameter	value
αx	2500
βx	1
βy	2
kx	8
ky	10
dz	1

Table 4-2-6

parameter	value
αx	3500
βx	1
βy	2
kx	650
ky	600
dz	2

Fig 4-2-7 Multi IFFL with the optimal parameters

Team:Waseda/Model

Modeling

Zombie vs Samurai circuit

4.2 The basic mothod and results of our genetical circuit

Reference

Multi IFFL