MODEL

Introduction

Initially, we developed an Ordinary Differential Equation (ODE) model to simulate the effects of our engineered bacteria on the internal environment of individuals with high-purine diets. This model demonstrated the safety and efficacy of our engineered bacteria in humans. As the project progressed, we gradually realized that this ODE model simulating human environments could not only be used for forward-looking analysis to validate the bacteria's effectiveness. By creating digital models of human internal environments and solving the ODE model, we can determine how much engineered bacteria is required to maintain stable uric acid levels in individuals with high-purine diets. In other words, this approach helps us predict the optimal dosage needed for bacterial supplementation.

Gut Purine-Blood Uric Acid Dynamic Model

The intestinal purine-blood uric acid dynamic model is our underlying model, which contains three core modules:

equation of intestinal purine:

$$\frac{dP}{dt}=I(t)-k_1\cdot \frac{B(t)}{B_{\text{ref}}}\cdot P(t)-k_2\cdot P(t)$$

$$I(t)=\frac{I_0}{\sqrt{2\pi }\sigma }\cdot e^{-\frac{(t-t_0{)}^2}{2{\sigma }^2}}$$

P (t) : Intestinal purine concentration

I (t) : A transient input function of purine simulated by a Gaussian pulse

k₁ : bacterial transport rate constant

k₂ : Rate constant for conversion of purine to uric acid

I₀ : Total purine intake in a meal

σ : time dispersion and width of purine absorption after feeding

The transient variable of purine in the gut is equal to the amount of purine taken from food minus the amount of purine transported by bacteria minus the amount of purine converted to uric acid

Dynamic equation of blood uric acid:

$$\frac{dU}{dt}=\alpha \cdot k_2\cdot P(t)-U(t)\cdot [\beta +K_{\text{reabs}}\cdot (1-\frac{K_{\text{reabs}}}{K_{\text{reabs}}+\gamma \cdot \frac{B(t)}{B_{\text{ref}}}})]$$

U (t) : serum uric acid concentration

α : Purine to uric acid ratio coefficient

β : renal excretion rate

γ : coefficient of bacterial ability to degrade uric acid

K_reabs : intestinal reabsorption coefficient

The instantaneous variable of uric acid in blood is equal to the amount of purine converted into uric acid in the intestine minus the amount of uric acid metabolized by the kidney minus the amount of uric acid reabsorbed by the intestine

equation of bacterial quantity:

$$B(t)=B_0\cdot e^{-kt}$$

B₀: Total amount of bacteria ingested at one time

k: natural decay coefficient of engineering bacteria

The engineered bacteria cannot proliferate in the intestinal environment. At time t=0, the engineered bacteria with a quantity of B₀ are introduced into the intestine, and the equation exists. Assuming that the natural decay rate of the bacterial population is proportional to the current bacterial population, with the proportionality constant being k, then:

$$\frac{dB}{dt}=-k\cdot B(t)$$

The exponential decay takes the form:

$$B(t)=B_0\cdot e^{-kt}$$

Fit of purine intake

We tried three different approaches to fit the intake of purines.

1. Consider I₀ as a transient pulse input and divide it into two stages:

(1)When t = 0, a direct intake of purine content I₀ occurs, and the equation exists as follows:

$$P(0^{+})=P(0^{-})+I_0$$

(2) Maintain no input to the intestinal tract:

$$\frac{dP}{dx}=-(k_1\cdot \frac{B(t)}{B_{\text{ref}}}+k_2)\cdot P(t)$$

Thus,we have

$$P(t)=I_0\cdot e^{-(k_1\cdot \frac{B(t)}{B_{\text{ref}}}+k_2)t}$$

The purine input is modeled as a single pulse at time t=0, with no subsequent input. This approach offers the advantage of a simple and intuitive model that is easy to solve, effectively demonstrating the decay trend after a large initial intake of purines into the intestine. However, it fails to align with the continuous intake process observed in actual diets, making it unsuitable for simulating multiple meals or sustained consumption scenarios, thus limiting its applicability

2. Consider I₀ as a continuous flow and model I (t) as a short time square wave:

$$I(t)=\{\begin{array}{ll}\frac{I_0}{\tau }& t_0\le t\le t_0+\tau \\ 0& t\le t_0,t\ge t_0+\tau \end{array}$$

The purine input is modeled as a constant continuous flow over specific time periods. This approach offers the advantage of more accurately replicating the phased nature of actual dietary intake (such as meal durations) compared to transient pulses, effectively capturing the characteristic pattern of sustained consumption within short timeframes. However, the abrupt changes in square waves differ from the gradual fluctuations in real-world purine intake during meals, resulting in less natural simulation of the consumption process

3. Consider I₀ as a continuous flow and model I(t) as a Gaussian pulse

$$\mathrm{FWHM}=\tau$$

$$\int I(t)=I_0$$

$$I(t)=\frac{I_0}{\sqrt{2\pi }\sigma }\cdot e^{-\frac{(t-t_0){}^2}{2{\sigma }^2}}$$

The Gaussian curve model for purine intake accurately replicates the gradual progression of food digestion and absorption. While its core strength lies in closely mirroring real physiological processes – effectively simulating the gradual increase and subsequent decrease of dietary purine intake – this approach presents notable limitations. The model's complexity and the technical challenges involved in parameter optimization make it more practical for clinical applications, ultimately justifying its adoption as the preferred solution.

Finally, the intestinal purine dynamic equation helps us describe the variation of purine content in the intestine.

In modeling Gaussian pulse inputs, σ represents the time dispersion width (standard deviation) of purine absorption after each meal, measured in hours. This value varies depending on the duration of the meal. Similarly, the peaks of the Gaussian curves simulating three meals differ as follows:

Meal	Peak time t0 (h)	Amplitude coefficient ci (mg)	Gaussian width σi (h)
breakfast	8	50 × (W/70)	0.5
lunch	13	80 × (W/70)	0.6
supper	19	60 × (W/70)	0.7

Under the two physiological conditions of considering the duration of feeding and intestinal emptying time, we take σ = 0.6 h for the model.

Derivation of dynamic equation of blood uric acid

At the beginning of modeling, our engineered bacteria had no design for uric acid transport. Therefore, the dynamic equation of blood uric acid initially only considered the amount of purine converted into uric acid in the gut and the amount of uric acid metabolized by the kidney.

$$\frac{dU}{dt}=\alpha \cdot k_2\cdot P(t)-\beta \cdot U(t)$$

The original equation neglected the critical physiological process of reabsorption. However, subsequent engineering modifications incorporated uric acid transport mechanisms in the gut microbiota. This advancement enabled a comprehensive reassessment of the initial model. To fully characterize the effects of probiotic supplementation, a new state variable U was introduced to accurately depict the metabolic processes.

$$\frac{d{U}_{\text{gut}}}{dt}=K_{\text{reabs}}\cdot (U-U_{\text{gut}})-r\cdot \frac{B(t)}{B_{\text{ref}}}\cdot U(t)$$

The transient variable of uric acid in the intestine is the amount of uric acid reabsorbed by bacteria and degraded by bacteria. The ability of bacteria to degrade uric acid is affected by bacterial density, so a restriction function should be added

The reabsorption component of serum uric acid should also be considered:

$$\frac{dU}{dt}=\alpha \cdot k_2\cdot P(t)-\beta \cdot U(t)-K_{\text{reabs}}\cdot (U-U_{\text{gut}})$$

Then, we combined the two equations and eliminated the variable U to get the final dynamic equation of blood uric acid:

$$\frac{dU}{dt}=\alpha \cdot k_2\cdot P(t)-U(t)\cdot [\beta +K_{\text{reabs}}\cdot (1-\frac{K_{\text{reabs}}}{K_{\text{reabs}}+\gamma \cdot \frac{B(t)}{B_{\text{ref}}}}\left)\right]$$

eGFR-Based Reabsorption Rate Estimation Model

The renal reabsorption rate (RR) of uric acid is a key parameter in maintaining blood uric acid homeostasis. While clinical measurements require direct assessment to obtain accurate values, the CKD-EPI equation can be utilized during modeling stages. This approach estimates estimated glomerular filtration rate (eGFR) based on age, gender, body weight, and serum creatinine levels, then converts it into RR for non-invasive, real-time personalized adjustments.

CKD-EPI equation:

$$\mathrm{e}\mathrm{G}\mathrm{F}\mathrm{R}=141\cdot (\mathrm{S}\mathrm{c}\mathrm{r}/k{)}^{\alpha }\cdot {0.993}^{\mathrm{a}\mathrm{g}\mathrm{e}}\cdot (1.018\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{e})$$

RR mapping formula:

$$\mathrm{R}\mathrm{R}=\{\begin{array}{ll}0.92& \mathrm{e}\mathrm{G}\mathrm{F}\mathrm{R}\ge 60\\ 0.92-0.03\cdot {\displaystyle \frac{60-\mathrm{e}\mathrm{G}\mathrm{F}\mathrm{R}}{10}}& \mathrm{e}\mathrm{G}\mathrm{F}\mathrm{R}<60\end{array}$$

symbol	meaning	unit
Scr	serum creatinine	mg dL-1
age	age	year
sex	sex	No unit
eGFR	Estimate glomerular filtration rate	mL min-1 1.73 m-2
RR	Intestinal reabsorption coefficient of urea	dimensionless

In clinical practice, Scr needs to be measured directly to obtain an accurate value. Here, we take 0.8 mg dL^-1 for calculation, and finally realize the calculation of K_reabs

Standard bacterial concentration curve

The wet experiment provided us with the speed of transporting purine in vitro at different concentrations of engineered bacteria.

The standard bacterial concentration curve was constructed by fitting the experimental data of wet in vitro transformation with Mie's equation.

$$\text{Michealis equation}:V=\frac{V_{\text{max}}\cdot \left[S\right]}{K_m+\left[S\right]}$$


$$V_{\max }=104.95,K_m=2.02e+8$$

$$V=\frac{104.95\cdot \left[S\right]}{2.02e+8+\left[S\right]}$$

Thus, the speed of purine transport by different engineering bacteria under certain conditions was obtained.

Uriguard: Model-Voice Assistant

Building upon our established model framework and parameter optimization, we aim to transform the intestinal purine-to-serum uric acid dynamic model into a practical, user-friendly tool. Through a three-phase process of model integration, hardware adaptation, and functional debugging, we have developed UriGuard – an interactive voice assistant that enables this advanced model to effectively serve patients with hyperuricemia.

1.Model Embedding

We converted the mathematical equations of the "Gut Purine-Serum Uric Acid Dynamic Model" into executable code modules and embedded them into the core program of UriGuard. Simultaneously, we connected with the official purine database from China Food Network. Through natural language processing technology, user-specified food information (such as "a bowl of pork rib soup") was automatically converted into purine intake data (e.g., 200mg) and transmitted in real-time to the model for computation. By combining patients' daily purine intake (calculated through diet) with personal characteristics (age, gender, weight), we automatically generated personalized daily intake amounts of probiotic medications (e.g., YES302/YES303).

Figure 4 A 45-year-old man weighing 70 kg ate 200mg of purine for breakfast

2.Hardware Adaptation

We built the firmware part of Uriguard by taking into account cost and performance, using common electronic components on the market

We selected the ESP32-S3-N16R8 development board (8 MB PSRAM + 16 MB Flash) to support lightweight model operations at the 70MB scale while ensuring system stability and future upgrade potential. For audio configuration, we integrated an INMP441 microphone with a MAX98357A digital amplifier: the former enables wide-field sound pickup, while the latter allows volume adjustment. This dual-implementation design ensures Uriguard's adaptability for both quiet hospital environments and noisy outdoor settings. Considering that hyperuricemia patients are predominantly elderly individuals with hearing impairments, we equipped the device with a 1.54ʺ IPS color display. This ergonomic solution enables clear text visibility for users with hearing difficulties and presbyopia, effectively addressing accessibility challenges in clinical environments.

In addition to this, we've listed more available hardware and their assembly methods, such as the ML307R 4G module. Teams can refer to our guide to select different modules according to their needs and assemble customized assistants. For detailed information, please consult our hardware section.

3.Function Debugging

We then performed functional debugging on Uriguard to ensure that the full link accuracy, latency, and robustness of the "ODE core" to "hardware" to "user" met the landing requirements.

We selected 10 groups of virtual users with different ages, genders, weights, and purine intakes. The user ranges covered standard youth, standard middle-aged, standard elderly, overweight, underweight, and kidney failure. We input them into Uriguard and direct solution respectively, and the average error was only 1.1% (see Table 2).

We have added fuzzy matching to NLP to accommodate user accents and dialects.

In addition, we have completed the adaptation of the multilingual library, so that it is more convenient for the iGEM team to reference and use our project.

Road Ahead

In the future, the accuracy of the model prediction can be verified by combining the intestinal pump in hardware.

The parameters in the model still need optimization. In different countries, regions, climates, and ethnic groups worldwide, dietary habits may cause variations in some parameters within the model. At this stage, for computational convenience, we focus on Chinese people in China. Subsequently, more extensive and diverse data should be applied to further refine individual parameter differences.

UriGuard holds significant expansion potential. With hardware open-source capabilities, the only remaining requirement is developing mathematical models that account for the internal environmental impacts of additional engineered bacteria. In the future, as more bacterial engineering models emerge within the iGEM community, UriGuard could be extended to large-scale health management systems and multi-strain collaborative systems, enabling adaptation to more complex diseases and scenarios.

Furthermore, UriGuard can evolve into a software solution by developing web or App-based interactive tools that automate the entire workflow—from user input and model calculations to dosage recommendations. Combined with our software——probiotics, we aim to pioneer efforts to break down existing biases against engineered probiotics.

Appendix

Table 1 Coefficients in the formula

symbol	Name (physical meaning)	unit	remarks
P(t)	Intestinal purine concentration	Mg/L-1	state variable
U(t)	Blood uric acid concentration	μg/mL-1	state variable
Ugut(t)	Uric acid concentration in the gut	μg/mL-1	intermediate variable
B(t)	Bacterial count of engineering bacteria	CFU/mL-1	state variable
I0	Total amount of purines ingested in a single intake	mg	Integral area of pulse input
I(t)	Purine intake rate function	Mg/h-1	Gaussian pulse form fitting
k1	Purine transport rate constant of bacteria	h-1
k2	Purine to uric acid conversion rate constant	h-1
k	Natural decay constant of bacteria	h-1
α	Protein conversion ratio coefficient of uric acid	dimensionless
β	Basal excretion rate of renal uric acid	h-1
γ	Coefficient of bacterial ability to degrade uric acid	h-1
Kreabs	Intestinal uric acid reabsorption coefficient	h-1
Bref	Reference bacterial concentration	CFU mL-1	data fitting
Bmax	Maximum load of bacteria in the gut	CFU	Safety limits
B0	Initial intake of bacteria	CFU	Dosage Design Output
eGFR	Estimate glomerular filtration rate	mL min-1 1.73 m-2	formula calculator
W	weight	kg	User input
A	age	year	User input
G	sex	No unit	Male = 0, female = 1 (for correction)

Table 2 Verification of ODE calculation error

ID	Year / y	G	W / kg	I0 / mg	True/ CFU×108	UriGuard / CFU×108	Error/ %
01	25	man	60	300	3.24	3.19	1.5
02	65	woman	55	180	1.76	1.75	0.6
03	45	man	85	400	4.55	4.61	1.3
04	50	woman	50	150	1.42	1.44	1.4
05	30	man	70	250	2.83	2.80	1.1
06	70	man	75	220	2.48	2.52	1.6
07	35	woman	65	280	2.51	2.49	0.8
08	55	man	90	350	3.97	4.02	1.3
09	60	woman	58	200	1.91	1.89	1.0
10	48	man	68	260	2.94	2.91	1.0