5-Fluorocytosine (5-FC) can be converted to 5-Fluorouracil (5-FU) which is a chemotherapy drug. Genetically engineered E. coli containing cytosine deaminase can convert 5-FC to 5-FU [1]. The dynamics of this conversion and 5-FU's tumor-killing effect need to be modeled to optimize the cancer treatment.

The purpose is to develop a mathematical model to describe:(1) The pharmacokinetics of converting 5-FC to 5-FU by E. coli.(2) The pharmacodynamics of 5-FU in inhibiting tumor growth.The model can help optimize the engineered E. coli and 5-FC dosage for effective tumor treatment.

5-FC and 5-FU transports into or out of E. coli following first-order kinetics[2]. Cytosine deaminase follows Michaelis-Menten kinetics in converting 5-FC to 5-FU[3]. Tumor cells follow exponential growth with inhibition by 5-FU following Emax model[4].

Equation 1
dFC = km_t*FC_ex - Vmax*FC/(Km + FC)*E0 This describes the rate of change of 5-FC concentration inside bacteria. First term is 5-FC transport into bacteria, and second term is 5-FC conversion to 5-FU by enzyme reaction.Equation 2
dFU = Vmax*FC/(Km + FC)*E0 - k_ext*FUThis describes the rate of change of 5-FU inside bacteria. First term is 5-FU generation from 5-FC, and second term is 5-FU transport out of bacteria.Equation 3
dFU_ex = k_ext*FU - k_clr*FU_exThis describes the rate of change of 5-FU outside bacteria. First term is 5-FU transport out of bacteria, and second term is 5-FU_ex degration.By combining these differential equation models, we can simulate the full dynamics of 5-FC/5-FU conversion and transport process.

5-FU's tumor killing effect Equation：
E = Emax * C / (EC50 + C)Where:E: killing effect of 5-FU on tumor cellsEmax: maximum killing effect of 5-FUC: concentration of 5-FUEC50: concentration of 5-FU that produces half EmaxThis is a typical Emax equation describing the concentration-effect relationship. When C is 0, E is 0. As C increases, E increases and approaches Emax. EC50 is the concentration at which E is 50% of Emax.The higher the C, the closer E gets to Emax.Tumor Growth Equation:
dN/dt = rN(1-N/K)-E*NWhere:N: number of tumor cellsr: tumor growth rateK: carrying capacityE: killing effect from 5-FUExplanation:First term describes logistic tumor growth, second term incorporates 5-FU killing effect, combining the two terms gives tumor growth dynamics with 5-FU inhibition.

The test results of the model are consistent with the experimental data. Under the condition of 5mM 5-FU_ex, the survival rate of tumor cells is 32.4%(Figure 1). So this model well describes the conversion of 5-FU and inhibition effect on tumor over time.

The model provides a basis for quantitative optimization of this tumor suppression, which establishes a pharmacokinetic-pharmacodynamic relation for the 5-FC/5-FU system. More complex models can be explored, e.g. adding cell uptake kinetics. What’s more, potential resistance development could be incorporated.

5-FC and genetically engineered E. coli can convert 5-FC to 5-FU and kill tumor cells in the gastrointestinal tract. To optimize this tumor-targeting system, we need a model of bacteria movement, aggregation, and killing effect.

The purpose is to develop a mathematical model to describe: (1) The movement and accumulation of engineered bacteria in the gut. (2) Bacteria aggregation around tumor sites. (3) Bacteria-mediated killing of tumor cells. This model can help optimize the tumor-targeting system.

The model makes the following assumptions: The gut is simplified as a 1D tube. Tumor cells distribute at specific sites in the gut tube. Bacteria move by passive circulation and chemotaxis towards tumor [5].

The modeling approach contains: Model gut shape and tumor distribution, establish bacteria movement by advection-diffusion-chemotaxis equations:dB/dt = D*B – v*B + χ(B * C)Where:B - bacteria concentrationD - diffusion coefficientv - flow velocityχ - chemotaxis coefficientC - concentration of chemoattractants from tumorCouple with 5-FU effect model on tumor cells:E = Emax * C / (EC50 + C)dN/dt = rN(1-N/K)-E*N

We show the simulate bacteria spatial distribution at 12h, E. coli has spread near tumor cells, The specific distribution is shown in the Figure 2.

By calculating the spread of E. coli and 5-FC, and the previously established 5-FU transformation and 5-FU killing models on tumor cells, we calculated the concentration of 5-FU_ex at the tumor cell location and the growth curve of tumor cells. After the above calculations, 5-FC of 116.2 mM/L can completely kill tumor cells after diffusion and transformation, and the Figure 3 shows the calculation results.

We established a mathematical model of genetically modified E. coli moving, aggregating, and killing tumor cells, validated with our experimental data, and ultimately used the model for medication guidance. Future work can include: Incorporate more complex gut structure, add detailed bacteria chemotaxis behavior. Extend model to multiple tumor sites, provide quantitative guidance for bacteria and 5-FC based cancer therapy.

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