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Model Design

Cyclebow Hybrid-Color Model

      Here we simulate the Cyclebow Hybrid-Color Model extended from the pure color model with RGB color combination. In order to simplify the model and minimize the number of regulatory molecules related to this model, we adopted the concept that we are able to control the effectiveness of the enzyme Vika and Vcre and then built up the hybrid-color changing process[15].

Here are our assumptions:

      1. The effectiveness of the enzyme is constant as long as the ratio of the concentration of the enzyme and the substrate is constant, which allows us to focus only on the effectiveness setting of each enzyme.

      2. Each plasmid that contains a single Cyclebow system is relatively independent, and the enzyme expressed by one plasmid are less likely to diffuse into other parts of the cell and affect the effectiveness of the enzyme.

      Based on these assumptions, we established the color combination model which is able to extend the colors to up to 7 generations, in which each color can be easily distinguished by human eyes. Parameters of the effectiveness of each enzyme is trained and the most distinguishable one is selected. The effectiveness of Vika and Vcre are set to be 50% and 40%, respectively.

HSC Cell Survival Model

      We are trying to explain the HSC reserve change after a certain amount of time. Related articles suggest that human HSCs replicate on average once per 40 weeks, and that when the total number of HSC exceeds the upper limit K (K=11,000), only one offspring cell survive[16].


      1. Each HSC can divide, emigrate to differentiate and die.

      2. Once differentiate, the cell will exhaust in a finite period of time.

      3. Each HSC is independent and have no connection with its neighbors.

      4. Different factors that cause different HSC behaviors are independent.

      We denote X0 to be the initial state of the total HSC implanted into the recipient; Xt to be the number of HSC reserve in the time of t; d to be the dividing rate; v to be the emigration rate, a to be the death rate; K to be the upper limit of HSC cell reserve. One previous paper suggests us to use the optimal parameter combinations, where d = 1/40 (divide once per 40 weeks[17]), v = 0.71d, a = 0.14d, X0 = 400, K=11,000[18].

Deterministic Model

      We developed a time sequence model that reflects the HSC number’s change before reaching the upper limit K.

x(t+dt) = x(t) + d * x(t) - v * x(t)
x(t0) = 400
x(t) < 11,000

      With the deterministic model, we can model the process of immigrating cells that serve as repairing cells and then exhausts after a certain period of time. In order to model this process, we adopted the previous generated number of emigration cells (# of emigration cell in each week) and introduced a new parameter as exhausting rate u. u is suggested to be 1/10 (one per 10 weeks), and is proved suitable for human HSCs。

y(t+dt) =y(t) - y(t) * u(t) + v * x(t)
y(t0) = 0

Stochastic Model

      According to one related modelling paper <The replication rate of human hematopoietic stem cell in vivo>, the authors used a stochastic model where assumptions were made about HSC acting based on their unique intrinsic and micro-environmental signals and that these fates are independent[13].

      Besides our deterministic model, we also want to compare the performance between our deterministic model and stochastic one. We modified their equations to fit our situation and the equation as follow:

X(t+dt) = X(t) + 1, withprobabilityX(t)λdividedt
X(t+dt) = X(t) - 1, withprobabilityX(t)αdeathdt
X(t+dt) = X(t) - 1, withprobabilityX(t)νemigratedt
X(t0) = 400
X(t) < 11000

      We set up the starting point at 400 HSCs and run the stochastic function to simulate HSC cell growth till it reaches its maximum capacity, which is 11,000.