the Dynamic of Hp-SHEEP-Human System

We have already found that Hp-Human system have two stable states—i-state and p-state. For i-state, Hp brings no harm but good to humans. Unfortunately, i-state is semi-stable and risk of it transmitting into p-state is unneglectable. Diet helps to stabilize i-state but it is hard to follow and its effect is limited. Here we designed SHEEP to form a homeostatic state between Hp, human, and our engineered bacteria. The model below explains how we transform a semi-stable state into a permanent stable state.

Part 1. the Master Equation

By adding the engineered bacteria into the equation describing the Hp-Human system, we get the master equation of our model.

Parameters used are listed below

Variable	Meaning	Dimension
$$L_M$$	Population of LA in mucus	b
$$L_A$$	Population of LA attached	b
$$P$$	Concentration of drugs	P
$$H_B$$	Population of coccoid HP	b

Table 1. Variables Involved in the Model

Variable	Meaning	Value	Dimension
$$N_L$$	Nutrient dependent constant of LA activity	0.45	N
$$r_L$$	Maximum growth rate of LA	10	d^-1
$$\alpha_L$$	Attachment rate of LA	0.4	d^-1
$$\beta_L$$	Dissociation rate of LA	1	d^-1
$$V_L$$	Maximum kill speed to LA	20	d^-1
$$P_L$$	LA's drug sensitivity index	1.4	P
$$U$$	Maximum lysis rate of LA	0.5	d^-1
$$H_0$$	Density index of dissociated HP	100	b
$$m$$	Hill coefficient	1.2	1
$$K$$	Average drug production amount	0.1	P/b
$$a_L$$	Extra nutrition for breeding of LA	$$N_L/50$$	N/b
$$b_L$$	Nutrition for growth of LA	$$N_L/100$$	N/(bd)
$$V_H$$	Maximum kill speed to HP	20	d^-1
$$P_H$$	HP's drug sensitivity index	1	P
$$n$$	Hill coefficients of drug	5	1
$$\sigma$$	Kill disability index towards coccoid HP	1	1
$$\lambda$$	Sensitivity decline index of coccoid HP	0.01	1
$$\chi$$	Maximum conversion speed of HP	1	d^-1
$$N_B$$	Index of nutrition deficiency of HP conversion	$$N_H/100$$	N
$$\omega$$	Index of conversion sensitivity of HP in drugs	0.05	1
$$\kappa$$	Average drug amount to kill	0.05	P/b
$$\mu_B$$	Natural mortality of Coccoid HP	0.003	d^-1

Table 2. Parameters Involved in the Model

Assuming that the lysis of our engineered bacteria when meeting H.pylori (Hp) as well as death caused by LL-37 can be both described by Hill equation, we write the equations for engineered bacteria

$$ \begin{align} \dfrac{dL_M}{dt}&=g_Mr_L\dfrac{N}{N+N_L}L_M-\mu_MH_M-\alpha_L\left(1-\dfrac{H_A+H_B+L_A}{A}\right)L_M+\beta_LL_A-\dfrac{V_L}{1+(\frac{P_L}{P})^n}H_M-\dfrac{U}{1+(\frac{H_0}{H_M+H_A})^m}L_M\tag{8}\\ \dfrac{dL_A}{dt}&=g_Ar_L\dfrac{N}{N+N_L}L_A-\mu_AL_A+\alpha_L\left(1-\dfrac{H_A+H_B+L_A}{A}\right)L_M-\beta_LL_A-\dfrac{V_L}{1+(\frac{P_L}{P})^n}L_A-\dfrac{U}{1+(\frac{H_0}{H_M+H_A})^m}L_A\tag{9}\\ \end{align} $$

We further assume that the production of LL-37 antimicrobial peptide is proportional to the amount of died La

$$ K\left(\dfrac{V_L}{1+(\frac{P_L}{P})^n}+\dfrac{UL}{1+(\frac{H_0}{H_M+H_A})^m}\right)(L_M+L_A) $$

Together with LL-37’s consumption, the complete equation of LL-37 is rewritten as

$$ \begin{align} \dfrac{dP}{dt}=K\left(\dfrac{V_L}{1+(\frac{P_L}{P})^n}+\dfrac{UL}{1+(\frac{H_0}{H_M+H_A})^m}\right)(L_M+L_A)-\kappa\left(\dfrac{V_H}{1+(\frac{P_H}{P})^n}(H_M+H_A)+\dfrac{1}{\lambda}\dfrac{\sigma V_H}{1+(\frac{P_H}{\lambda P})^n}H_B\right)+D_z(f(t)-P)\tag{7} \end{align} $$

Taking the nutrient consumption of engineered bacteria into account, we rewrite the equation for N

$$ \begin{align} \dfrac{dN}{dt}=k_N\dfrac{I_N}{I_N+I}\dfrac{E^\xi}{E^\xi+E_N^\xi}-a_Hr_H\dfrac{N}{N+N_H}(g_MH_M+g_AH_A)-a_Lr_L\dfrac{N}{N+N_L}(g_ML_M+g_AL_A)-b_H(H_M+H_A)-b_L(L_M+L_A)+D_Z(N_0(t)-N)\tag{3} \end{align} $$

Part 2. One Cell Simulation

We run the simulation under different boundary conditions. The main results we obtain are listed below.

(a) Wild La has no advantage over Hp. Introducing a tiny amount of Hp into the already stabled state of pure La would result in the extinction of La.

(b) Engineered La could establish a stable state together with Hp. This state has even lower level of inflammation and immune system activation when comparing to i-state. Such stable state can be approached in a wide range of initial condition. However, if Hp is already in i-state or p-state, even engineered La have no chance.

(c) The stable state mentioned in (b) is robust to the change of environmental nutrition. Even doubling the environmental nutrition would not break the balance (the balance would only take a reversible move). We name the state as h-state (homeostatic state).

(a) emphasizes the necessity of engineering the La. Hp possess a huge advantage over other bacteria in the stomach environment, making it almost undefeatable in the competition for ecological niche. Even our engineered bacteria do not own an absolute advantage over Hp. (b) implies that SHEEP should not directly use engineered bacteria as a therapy, it is more rational to use traditional treatment to control the amount of Hp and then introduce engineered La. (c) evidenced that the stable state we obtain is a homeostasis. With SHEEP we no longer need to worry about the state transition and the following diseases. The results demonstrate that our design is successful.

Part 3. Model with Space Dimension

In previous simulation we get a satisfying result that SHEEP could establish and hold a homeostasis. However, such result is obtained in a one cell model while the stomach wall is more of a two-dimensional manifold. It is possible that the topological structure involved would influence the result. A factor related to topology is the clustering of bacteria. In models with non-zero space dimension, bacteria could have space distribution. If Hp somehow clustered, they may form a local i-state or p-state and our engineered La would loss its advantage. Simulations show that such worry does exist.

To run the simulation with space dimension, the master equation should be first transformed. In fact, all we have to do is to replace the time derivative operator $ \frac{d}{d t} $ with the operator for diffusion $ \partial_{t}-D_{i} \nabla^{2} $, in which D_i is the diffusion constant of component i.

Variable	Meaning	Value	Dimension
$$D_L$$	Diffusion rate of LA in mucus	4	X²/d
$$D_H$$	Diffusion rate of HP in mucus	4	X²/d
$$D$$	Diffusion rate of general macromolecules in mucus	0.4	X²/d

Table 3. Diffusion constant Involved in the Model

For one-dimensional simulation, the operator transformation is $\frac{d}{d t} \rightarrow \partial_{t}-D_{i} \partial_{x}^{2}$. We run the simulation with reflective boundary conditions and assume that Hp forms a local i-state. Results of the simulation show that the clustering Hp step by step take over the places and engineered La extinct.

A two-dimensional simulation is also done. In the simulation, we assume that the system evolves with rotational symmetry, therefore the space location can be described with one single variable r. the operator transformation in this case is $\frac{d}{d t} \rightarrow \partial_{t}-D_{i}\left(\partial_{r}^{2}+\frac{1}{r} \partial_{r}\right)$. We continue to use reflective boundary conditions and we assume that Hp form a local i-state on a centered disc. Simulation results differ with different radiuses of the disc. A critical radius is found to be around 10 X. The spread of clustering Hp is controllable if and only if the radius of the disc is smaller than the critical radius.