# Protein-Based System Models

## Introduction

We developed a number of models for the protein based system to simulate its dynamics. We used this model to inform us of the effectiveness of using a positive feedback loop, the difference between using a singly-inhibited and doubly-inhibited coil, and also the possibility of using different types of coiled-coils to improve the performance of our system.

When designing this system, we were aware of the potential of more false positives, as the inhibitory coil could unfold without being cleaved by Cruzipain. This allows the two coiled-coils that are now uninhibited to associate and the resulting split TEV protease might activate the system. It is our goal to quantify this effect.

## Methodology

The model simulates the cleavage by Cruzipain using Michaelis-Menten kinetics, including the competitive inhibition of multiple identical substrates. The association and dissociation rates of the coiled-coils are estimated from the dissociation constant. The split TEV protease fragments are assumed to have no effect on the association and dissociation of the coiled coils.

The following parameters were used:

Parameter Variable Name Value Reference*
Catalysed rate of reaction for Cruzipain $$k_{cat,c}$$ $$10.8$$ A
Michaelis Constant for Cruzipain $$K_{m,c}$$ $$5.8*10^{-6}$$ A
Catalysed rate of reaction for splitTEV $$k_{cat,sTEV}$$ $$0.222$$ C & D
Michaelis Constant for splitTEV $$K_{m,sTEV}$$ $$121*10^{-6}$$ C & D
Change in Free Energy of the Dimerisation of A and B $$\Delta G_{A:B}$$ $$-6.1$$ B
Dissociation Constant of A coil and B coil $$K_{d,A:B}$$ $$e^{\frac{\Delta G_{A:B}}{R*T}}=e^{\frac{-6.1}{1.99*10^{-3}*(273+25)}}=0.0038$$ B
Dissociation Rate of A coil and B coil $$k_{d,A:B}$$ $$K_{d,A:B}*k_{a,A:B}=3.8*10^{-7}$$ B
Association Rate of A coil and B coil $$k_{a,A:B}$$ $$10^4$$ B
Empirical Adjusted Ratio for A:Inhibitory Coil and B Dissociation Constant based on Experimental Data $$R_{A:B^*,B}$$ $$32$$ B
Empirical Adjusted Ratio for A and A:B Dissociation Constant based on Experimental Data $$R_{A,A:B}$$ $$17$$ B
Empirical Adjusted Ratio for A:Inhibitory Coil and A:B Dissociation Constant based on Experimental Data $$R_{A:B^*,A:B}$$ $$1000$$ B

The following species were used

Species Name Symbol Used
Cruzipain $$Cruzipain$$
N Terminal SplitTEV A B* $$V300$$
N Terminal SplitTEV A B* with B* cleaved $$V300_{cleaved}$$
C Terminal SplitTEV B $$V400$$
V200 anchored OMV (sterically-hindered) $$V200_{OMV}$$
V300 anchored OMV (sterically-hindered) $$V300_{OMV}$$
V400 anchored OMV (sterically-hindered) $$V400_{OMV}$$
Hirudin $$Hirudin$$
SplitTEV with the inhibitory coil is cleaved (V300_cleaved + V400) i $$splitTEV_{clean}$$
SplitTEV with unfolded inhibitory coil (V300 + V400) $$splitTEV_{coil}$$
The sume of splitTEV_clean + splitTEV_coil $$splitTEV$$
V200_OMV + V300_OMV + V400_OMV + V300 $$substrates_{cruzipain}$$
V200_OMV + V300_OMV + V400_OMV + V300 $$substrates_{splitTEV}$$

The following reactions were modelled for single inhibitory coil:

Cruzipain Cleavage Reactions

$$1. V200_{OMV} + Cruzipain \leftrightharpoons V200_{OMV}:Cruzipain \to Hirudin + Cruzipain$$ $$2. V300_{OMV} + Cruzipain \leftrightharpoons V300_{OMV}:Cruzipain \to V300 + Cruzipain$$ $$3. V400_{OMV} + Cruzipain \leftrightharpoons V400_{OMV}:Cruzipain \to V400 + Cruzipain$$ $$4. V300 + Cruzipain \leftrightharpoons V300:Cruzipain \to V300_{cleaved} + Cruzipain$$

splitTEV Cleavage Reactions

$$1. V200_{OMV} + splitTEV \leftrightharpoons V200_{OMV}:splitTEV \to Hirudin + splitTEV$$ $$2. V300_{OMV} + splitTEV \leftrightharpoons V300_{OMV}:splitTEV \to V300 + splitTEV$$ $$3. V400_{OMV} + splitTEV \leftrightharpoons V400_{OMV}:splitTEV \to V400 + splitTEV$$ $$4. V300 + splitTEV \leftrightharpoons V300:splitTEV \to V300_{cleaved} + splitTEV$$

Coiled-Coil Interactions

$$1. V300_{cleaved} + V400 \leftrightharpoons splitTEV_{clean}$$ $$2. V300 + V400 \leftrightharpoons splitTEV_{coil}$$

The following reactions were modelled for double inhibitory coil:

Cruzipain Cleavage Reactions

$$1. V200_{OMV} + Cruzipain \leftrightharpoons V200_{OMV}:Cruzipain \to Hirudin + Cruzipain$$ $$2. V300_{OMV} + Cruzipain \leftrightharpoons V300_{OMV}:Cruzipain \to V300 + Cruzipain$$ $$3. V400_{OMV} + Cruzipain \leftrightharpoons V400_{OMV}:Cruzipain \to V400 + Cruzipain$$ $$4. V300 + Cruzipain \leftrightharpoons V300:Cruzipain \to V300_{cleaved} + Cruzipain$$

splitTEV Cleavage Reactions

$$1. V200_{OMV} + splitTEV \leftrightharpoons V200_{OMV}:splitTEV \to Hirudin + splitTEV$$ $$2. V300_{OMV} + splitTEV \leftrightharpoons V300_{OMV}:splitTEV \to V300 + splitTEV$$ $$3. V400_{OMV} + splitTEV \leftrightharpoons V400_{OMV}:splitTEV \to V400 + splitTEV$$ $$4. V300 + splitTEV \leftrightharpoons V300:splitTEV \to V300_{cleaved} + splitTEV$$ $$5. V400 + splitTEV \leftrightharpoons V300:splitTEV \to V400_{cleaved} + splitTEV$$

Coiled-Coil Interactions

$$1. V300_{cleaved} + V400 \leftrightharpoons splitTEV_{v400coil}$$ $$2. V300 + V400 \leftrightharpoons splitTEV_{bothcoils}$$ $$3. V300 + V400_{cleaved} \leftrightharpoons splitTEV_{v300coil}$$ $$4. V300_{cleaved} + V400_{cleaved} \leftrightharpoons splitTEV_{clean}$$

These reactions were modelled in ODEs and simulated using the ode15s function at an absolute tolerance of 10-30 and a relative tolerance of 10-7.

## Single Inhibitory Coil or Double Inhibitory Coil

We wanted to know if using a single inhibitory coil system or a double inhibitory coil system would give the best result. The main benefit of a double inhibitory coil system is that it is less likely for both inhibitory coil to unfold and cause the coiled-coils to dimerise. As shown by Shekhawat et al., a doubly inhibited coiled-coil has a 1040 fold-difference in output in the absence and presence of a TEV protease, whereas a single inhibited coiled-coil only has a 22 fold-difference. Hence, we wanted to see how a double inhibitory coil system would affect the performance of a system.

The simulations of both models show that the single inhibitory system approaches the threshold (t = 10.42 min) much faster than the double inhibitory system (t = 55.89 min). This is as expected as the additional inhibitory coil in the double inhibitory coil system increases the total amount of substrates that the cruzipain has to cleave. Hence, the production of hirudin is much slower in the double inhibitory coil system and but accelerates later on. The single inhibitory coil system has a steady initial production rate and as a result approaches the threshold much faster.

To see how these systems affect the rate of false positives, we ran the simulations with 0 cruzipain. However, there was very little production of hirudin in either system, and is insufficient to inhibit blood coagulation. Hence, modelling showed that we should stay with a single inhibitory coil system.

In fact, we tried running it without any feedback loop, and it shows that the direct cleavage of an inactivated form of hirduin by cruzipain actually has the fastest threshold time.

## Stochastic Modelling – Intrinsic Noise

To assess how stochasticity might affect our system, we simulated the protein-based system using the tau-leaping method.

We plotted the final hirudin amount in a histogram and fitted a normal probability distribution function to the data. Looking at the horizontal scale, we see that the variation in the final amount of Hirudin is very low. In fact, the relative standard distribution is $$4.78*10^{-8}$$, indicating that the effects of inherent stochasticity is negligible in our system at a reaction volume of 30 µL.

Hence, this justifies our use of deterministic models in simulating both our DNA-based and Protein-based models.

## Discussions

We showed that the protein-based system works and that it's feedback loop indeed accelerates the release of hirudin. However, the coils we have chosen have a high Kd and that decreases the effectiveness of the feedback loop. As seen, the feedback loop increases the competitive inhibition of Cruzipain substrates, and so a system without feedback loops turned out to have the fastest threshold time.

## References

Index Reference
A Dos Reis, F. C. G. et al. (2006) ‘The substrate specificity of cruzipain 2, a cysteine protease isoform from Trypanosoma cruzi’, FEMS Microbiology Letters, 259(2), pp. 215–220. doi: 10.1111/j.1574-6968.2006.00267.x.
B Shekhawat, S.S., Porter, J.R., Sriprasad, A. and Ghosh, I., 2009. An autoinhibited coiled-coil design strategy for split-protein protease sensors. Journal of the American Chemical Society, 131(42), pp.15284-15290.
C Wehr, M.C., Laage, R., Bolz, U., Fischer, T.M., Grünewald, S., Scheek, S., Bach, A., Nave, K.A. and Rossner, M.J., 2006. Monitoring regulated protein-protein interactions using split TEV. Nature methods, 3(12), pp.985-993.
D Cabrita, L.D., Gilis, D., Robertson, A.L., Dehouck, Y., Rooman, M. and Bottomley, S.P., 2007. Enhancing the stability and solubility of TEV protease using in silico design. Protein science, 16(11), pp.2360-2367.