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Oscillator Modification Model

2022 WHU-China

Oscillator Modification Model

    Introduction

     To address the rejection of drugs in patients with depression, we modified Bacillus subtilis as the model of Bifidobacteria longum to produce the depression drug SAMe for a long time and periodically, which achieves the purpose of once-administered, long-term effectiveness and reduces the possibility of drug rejection by patients. Even though the oscillator module is included in the SimBiology model, the strong coupling with other modules makes it difficult to obtain insight into how to make the oscillation stable and how to regulate the oscillation period. What’s more, cell-to-cell interactions were also ignored. Since the oscillator element in the bifidobacteria plays a crucial role in our project, we perform a separate detailed modeling of the oscillator to present guidance to the experimental group in the process of designing the oscillator. The main principle is to use the direct cyclic inhibition of expression of the three proteins, which makes the three proteins change one after another, producing an oscillatory effect. Downstream of one of these proteins is SAMe synthetase opSam2. Thus along with the periodical expression of opSam2, the anti-depression drug, Sadenosyl methionine, is achieved, oscillating in quantity over time, for the purpose of cyclic drug administration. This model is to simulate the change of the drug produced by the colony with the oscillation of the oscillator and to illustrate the feasibility of our scheme. To a certain extent, it serves as a guide and expands the role of wet experiments.

    Symbols and definitions

     Below are definitions of some of the symbols that will be used by the model.
    Symbol     definition
t t     Time
i i     One of three genes
m i m_i     The i i th mRNA
p i p_i     The i i th protein
a i a_i     Constitutive transcription rate
b i b_i     Leaky or repressed transcription rate
K K     Switching concentration of repressors
n n     Cooperativity of repressors, n = 2 n=2 usually
δ i \delta_i     Degradation rate of mRNA
c i c_i     Translation rate of protein
μ \mu     The growth rate of cells
μ 0 \mu_0     The instantaneous growth rate of the cell
λ ˉ i \bar{\lambda}_i     Degradation rate of the ith protein
σ i \sigma_i     Variance of Gaussian distribution

    Model Assumptions

  •     To facilitate the model’s simulation of the combined oscillatory drug production of the entire colony, we assume that the colony is two-dimensional monolayers of tightly packed engineered bacteria, and their growth rates are isotropic;
  •     Due to the dramatic complexity of the situation in the human body, we assume that the oscillatory properties of the bacteria do not vary with the physicochemical factors in the human body to highlight the oscillatory properties of the flora.

    Model Construction and Explanation

     In our oscillator model, we consider not only the most basic processes of transcription and translation of proteins and the repression between the corresponding proteins. The growth situation of the engineered bacteria in the colony and the noise caused by uncertainties in promoter leakage and other environmental factors are also considered. Finally, we get the following model:
     We use the following set of differential equations to depict the oscillator model:
     In the following two sections, we will explain detailed process to build the above model.

    Growth Model

     Cell growth is influenced by the viscous drag between cells and cells and cells and substrates. The cells in such a system are constrained to a monolayer. We utilize an individual-based model to simulate cell growth over time. A simple model as follows can be introduced to take the job.
μ ˉ ( t ) = e − r ( t ) / r 0 ( 2 ) \bar{\mu}(t)=e^{-r(t)/r_0}\quad\quad\quad(2)
    where r 0 r_0 (8.23±1.69 cell diameters) is the characteristic length scale of the radial variation in growth rate. The expansion rate of the colony area can estimate the growth rate of the colony, and is given by the divergence of the velocity field,
∇ ⋅ v = 1 A d A d t ( 3 ) \nabla \cdot \boldsymbol{v}=\frac{1}{A}\frac{dA}{dt}\quad\quad\quad(3)
    where A A is the cell area and v v is the velocity. As assumption mentioned, growth is isotropic, we can decompose the expansion rate equally into its radial and perpendicular components. We set v v the velocity in the radial direction r r , and w w the velocity in the perpendiculat direction s s , equation above introduces,
d v d r + d w d s = 2 d v d r = μ ( r ) ( 4 ) \frac{dv}{dr}+\frac{dw}{ds}=2\frac{dv}{dr}=\mu(r)\quad\quad\quad(4)
     Given v = d r / d t v = dr/dt , we can rescale time as t → t μ 0 t → tμ_0 and radial distance as r → r 0 r → r_0 to get:
d d r ( d r d t ) = 1 2 e − r ( 5 ) \frac{d}{dr}\Big(\frac{dr}{dt}\Big)=\frac{1}{2}e^{-r}\quad\quad\quad(5)
     Integrating by r r and t t we obtain final result:
μ ˉ ( t ) = ( 1 + e x p ( t − τ 2 ) ) − 1 ( 6 ) \bar{\mu}(t)=\Big(1+exp\Big(\frac{t-\tau}{2}\Big)\Big)^{-1}\quad\quad\quad(6)
    where τ = − 2 l o g ( e x p ( r ( 0 ) ) − 1 ) τ = −2 log(exp(r(0)) − 1) , and r ( 0 ) r(0)

    the Process of Building Oscillator Model

     We apply a differential equation model coupled with growth model to track the amount of different proteins. We take two steps, transcription & translation, to simulate repressilator genetic circuit, which can be formulated as follows,
     Since mRNAs are typically short lived, we may assume quasi-steady state concentrations and set d m i d t = 0 \frac{dm_i}{dt} = 0 and we obtain the new system below,
     Rescaling protein concentration as p j → p j / K j p_j → p_j/K_j and time by t → t μ 0 t → tμ_0 with μ 0 μ_0 the maximal growth rate. Also we introduce Gaussian noise with variance σ i σ_i to simulate the effect of the real environment on the oscillator, donated as N ( σ i ) N(σ_i) ,thus obtaining final oscillator model for single cell:
    where α i = c i a i / δ i μ 0 K \alpha_i = c_ia_i/δ_iμ_0K and λ ˉ = λ / μ 0 \bar{\lambda} = λ/μ_0 . Then for a colony with radius R ( R = m a x ( r ( 0 ) ) ) R(R = max(r(0))) , integrating t t and r r , we can get protein produced by the colony until time t t , just as:
     The our final model is obtained and the three proteins produced in the colony region G G are as follows:

    Simulation Result and Conclusion

    Basic simulation

     Simulation shows, in Figure 1, the amplitude of the oscillations of the three proteins in a single cell decreases with time and stabilizes at the end. As Figure 2,3 shows, three protein contents alternately rise and fall, in line with the expected working effect of the oscillator. In Figure 4, we can see that as the oscillation proceeds, the three protein steps converge and the oscillatory collapse can be considered.
     Figure 1: Overall trend of protein content
     Figure 2: Early oscillation
     Figure 3: Late oscillation
     Figure 4: Oscillation collapse

    Influence of Degradation Rate of Protein

     We gradually increased the degradation rates of the three proteins ( λ ˉ i \bar{\lambda}_i ) to observe the changes in oscillatory behavior, especially the oscillation period as well as the time when oscillatory collapses. It is worth mentioning that we consider oscillatory collapse when the protein changes less than 20% of the mean value during the oscillation.
     The effect of the time of oscillatory breakdown with the rate of protein degradation can be seen in Figure 5: the two are negatively correlated.
     Figure 5: λ ˉ \bar{\lambda} -Time to collapse
     The variation of oscillation period with the rate of protein degradation is shown in Figure 6: The oscillation period decreases with increasing protein degradation rate, and even when λ ˉ \bar{\lambda} , the oscillation period decreases abruptly and the oscillation is nearly dissipated.
     Figure 6: λ ˉ \bar{\lambda} -Periodicity

    Influence of Colony Size

     We vary the colony size by changing the colony radius r r . We observed the oscillation characteristics at different colony sizes and obtained the relationship between r r and oscillation periodicity as shown in Figure 7. We found that a certain colony size has a stable oscillation periodicity, while smaller colonies oscillate unstably and with very short periodicity.
     Figure 7: r r -Periodicity

    Model Conclusion

     Based on basic results we can understand that our model is consistent with the real situation: the content of the three proteins oscillates and changes, while the amplitude of the oscillations diminishes with time and the oscillations collapse. In addition, we simulated the effect of protein degradation rate on the stability of oscillation and cycle length, and we found that low degradation rate is beneficial to the stability of oscillation, while low degradation rate is beneficial to the extension of cycle. Finally, we found that smaller colonies are not conducive to the stability of the oscillations, while a certain size of the colony facilitates oscillation stability. This also gives us some insight into the design of our project: we do not need to over-emphasize the drug expression of individual cells, but to achieve the target drug expression by expanding the number of colonies, which on the one hand reduces the design difficulty of the engineered bacteria, and at the same time enables the stability of the oscillation effect of the colonies.

    Reference

    [1] Yáñez Feliú, Guillermo, et al. "Novel tunable spatio-temporal patterns from a simple genetic oscillator circuit." Frontiers in bioengineering and biotechnology 8 (2020): 893.