Overview

In 2022, BUCT-IGEM has proposed a novel approach to hair care, that of using probiotics to consume scalp oil and produce caffeine for hair growth. The goal of our modelling is to gain insight into the project and predict the effect of the engineered bacteria.

Two models are developed to verify, predict and improve the feasibility of the design. The first model is a mathematical model to predict the variation in yield of caffeine production by engineered bacteria. The model is highly consistent with the experimental results and gives a good picture of the growth of the engineered bacteria and the variation in caffeine production, demonstrating the feasibility of our design. The second model, also mathematical, is used to predict the growth of the engineered bacteria in the scalp environment. The model provides a good indication of the trends in the respective populations of the engineered bacteria and Malassezia^[5] in competition for scalp oil, and provides us with the promoter density for designing the strain suicide mechanism.

Caffeine yield prediction

- to predict yield through product synthesis kinetics

The model is constructed to predict how the engineered bacterium works in practical use for caffeine production and to demonstrate that the bacterium can provide enough caffeine to promote hair growth.

1) Bacterial growth kinetic model

Logistic equations^[7] are the most used models to describe the kinetics of bacterial growth. First proposed by Verhulst in 1838, the Logistic model with a differential equation form is mainly used to describe the growth pattern of a population with limited resources. The differential form is as follows^[4]:

where: X- total bacterial concentration, denoted by OD600; t- growth time, d; μ- ratio growth rate.

The integral of the formula gives:

Based on the experimental data, the fit gives μ = 0.2495, X₀= 0.057, which can be substituted to give

The fitting result is as follows

It can be seen that the fit is good, with an R² of 0.85284, initially satisfying the actual growth of the engineered bacteria designed under non-competition.

2) Kinetic model of product synthesis

By comparing the experimentally obtained caffeine production curve with the change in the number of engineered bacteria, we have found that the bacteria are able to synthesize caffeine during both the growth and stabilization periods. To be more precise, the growth process starts with both caffeine production and an increase in numbers, and then for a certain period the bacterium does not reproduce, but the synthesis process still takes place. Therefore, we can say that the synthesis process of this product is a partial growth coupling type, using the Luedeking-Piret equation.

where: p-caffeine yield, % (mass fraction); m1- formation parameters of non-growth-related products; m2- growth related product formation parameters

The integral of the formula gives:

By fitting, m1=0.0027, m2=0.5011, which can be obtained by substituting them

And the fitting result:

It can be seen that the result is good, with an R² of 0.99781, and the model can be used initially for the prediction of caffeine yield.

According to the available literature, a caffeine concentration of less than 0.4 mg/L is sufficient to promote hair follicle growth^[1], whereas the maximum production from our experiments and the output predicted by the model is much higher than 0.4 mg/L. Considering the actual conditions of the scalp, such as caffeine penetration efficiency and cellular absorption rate, the amount of caffeine required to promote hair follicle growth would be much larger than the amount obtained in the literature. Therefore, our experimental data and the output predicted by the modelling would better match the requirements for the promotion of hair follicle growth.

Prediction of the regulation of strain competition

- prediction of the effect of Malassezia inhibition and design of the suicide mechanism

1) To predict the competition inhibition of strains

Fermentation kinetics studies the dynamic equilibrium of bacterial growth, substrate consumption and product production during the fermentation process and its changing laws. The difference with traditional population kinetics lies in the introduction of the concept of resource limitation. The study includes the balance of energy and mass in the growth of microorganisms, the interrelationship between the growth rate of the bacterium, the production rate and the consumption rate of the substrate during fermentation, and the influence of environmental factors on these three factors and the conditions affecting the reaction rate. Depending on how the microorganisms are cultured, the fermentation kinetics model includes three types: batch culture, continuous culture and replenishment batch culture.^[6]

Our experiments are of the batch culture. Batch culture is a process in which microorganisms are cultured in a closed system with limited resources added at once. When t=0, the microorganisms to be cultured are inoculated into the medium in the fermenter and then incubated under suitable conditions. During subsequent growth, nothing is added other than oxygen, defoamer and controlled pH.

The most common model for describing the relationship between growth rates and resources during microbial growth is the following:

where X is the number of the i population, S is the total amount of resources currently in the environment, μ_B ,μ_D is the relative birth and death rates of the population, Y_G is the population yield factor (the number of populations generated/the amount of resources consumed), which represents the resources consumed to provide population growth, and m is the population maintenance factor, which represents the amount of resources consumed to maintain the normal life activities of the population.

The basic equation of competition between two populations can be written as

The traditional approach usually assumes that the parameters in the above equations are constant or of a particular functional form of X⁽¹⁾, X⁽²⁾ and S. We assume that the ratio of the number of populations to the number of resources satisfies a particular functional relationship. Specifically, the population will adapt as it grows due to changes in external environmental resources. For computational purposes, we assume that the adaptive adjustment of the population arises mainly from changes in the birth rate, so that μ_B⁽ⁱ⁾ is considered a variable in equation (2), while the other covariates are considered constants.

It is assumed that both populations satisfy the ratio growth law, in which population 1 is the engineered bacteria, population 2 is Malassezia and S is the scalp oil content^[2]. For population X⁽¹⁾there is, introducing a new variable Y⁽¹⁾=X⁽¹⁾/S, we have X⁽¹⁾=SY⁽¹⁾, and the derivative at both ends with respect to time t yields the following equation

Substituting into (2), we can get:

For the population X⁽²⁾, repeating the same steps yields:

Substituting (4) and (5) into the third equation of equation (2), it can be obtained

Simplify the equation again:

When Y⁽¹⁾ and Y⁽²⁾ are known functions of t, this equation is a chi-square ordinary differential equation with respect to S. The general solution can be written as

Thus X⁽¹⁾ and X⁽²⁾ can be solved as

Since the integral of (9) contains both Y⁽¹⁾, Y⁽²⁾, it is generally difficult to find the exact analytical expression. Considering the fact that S₀ is usually relatively large and the values of Y⁽¹⁾,Y⁽²⁾ are small when the time t is relatively short, we have

In this case, Equation (9) can be simplified to the following approximate expression

In order to obtain the final result, it is necessary to give the expressions for the intrinsic growth lawY⁽¹⁾, Y⁽²⁾. We assume that Y⁽¹⁾, Y⁽²⁾ satisfy the logistic equation, so

where the initial valueY₀⁽ⁱ⁾=X₀⁽ⁱ⁾/S₀, r⁽ⁱ⁾ is the intrinsic growth rate of population X⁽ⁱ⁾and Y_m⁽ⁱ⁾ is the intrinsic accommodation of the environment for population X⁽ⁱ⁾.

Therefore,

Assume that the initial values of populations X⁽¹⁾, X⁽¹⁾ and S are:

X₀⁽¹⁾=0.1 X₀⁽²⁾=0.05、0.07、0.1 which simulates the concentration of Malassezia in different scalp environments,S₀=1 ,t=100h, r⁽ⁱ⁾=0.2/h, Y_m⁽¹⁾=1, Y_m⁽²⁾=1 , μ_D⁽ⁱ⁾=0.05/h,Y_G⁽ⁱ⁾=1,m⁽ⁱ⁾=0.1/h

From the curves shown, it can be seen that the growth rate and change trend of the number of engineered bacteria at different initial concentration of Malassezia differ, but all show good inhibition.

In fact, a lot of the inflammation that occurs on the scalp is caused by Malassezia when the sebum is in full bloom because it grows and forms a biofilm. As long as our probiotics are able to compete for free fatty acids in the application context, they are able to disrupt Malassezia biofilm formation in the process of inhibiting Malassezia.

As the initial number of Malassezia increases, its growth rate and number will become larger and the time to reach the maximum number will decrease. For the engineered bacteria, the pressure of competing for resources is higher, leading to a decrease in number, but still higher than the number of Malassezia, which can inhibit its growth well, proving the feasibility of our design. Among the predicted results, 250CFU/mm² was the density at which our E. coli Nissle 1917 could inhibit Malassezia on the scalp.

This is an easy number to achieve, because the concentration of bacteria in the liquid is already 10⁸ cells/mL when OD600=1. The surface area of the human scalp is approximately 470cm², and our Nissle is mixed with 4mL of fluid and applied to the scalp. After calculation, we found that as long as the concentration of bacteria meets 10⁷ cells/mL, it can reach 250CFU/mm² on the scalp.

2) Selection of inductive promoters

By reading a large amount of literature, we have discovered a library of QS variants with both high dynamic ranges and low leakiness. Based on the regulatory roles of a CRP-binding site and the lux box to −10 region within luxR-luxI intergenic sequence in controlling the lux-type QS promoters, by varying the numbers of the CRP-binding site and redesigning the lux box to −10 site sequence, more precise regulation of the strain density is possible.

In order to ensure biosafety, we desire to find an optimal density at which the strain could both suppress Malassezia with various concentrations well and simultaneously initiate the expression of the mazF gene, then induce apoptosis and ensure that the engineered bacteria don not overspread. It is clear from the graph fig.3 that for coping with different scalp conditions, the promoter density of the strain's suicide mechanism should be above 0.21 (OD600), ensuring that Malassezia with any density can be well suppressed.

Reference

1. Chen, Lei et al." Effect of caffeine on human hair follicle and dermal papilla cells cultured in vitro." Journal of Clinical Dermatology 40.06(2011):323-327.

2. Cui, Zixin et al. “Antifungal Effect of Antimicrobial Photodynamic Therapy Mediated by Haematoporphyrin Monomethyl Ether and Aloe Emodin on Malassezia furfur.” Frontiers in microbiology vol. 12 749106. 16 Nov. 2021, doi:10.3389/fmicb.2021.749106

3. Ge, Chang et al. “Redesigning regulatory components of quorum-sensing system for diverse metabolic control.” Nature communications vol. 13,1 2182. 21 Apr. 2022, doi:10.1038/s41467-022-29933-x

4. Xia, Lu et al." Study on static fermentation kinetics of cellulose producing strain from methanol." Biotechnology 21.02(2011):70-73.

5. Saunders, Charles W et al. “Malassezia fungi are specialized to live on skin and associated with dandruff, eczema, and other skin diseases.” PLoS pathogens vol. 8,6 (2012): e1002701. doi:10.1371/journal.ppat.1002701

6. Shang, Jieli. “Ratio- dependent competitive growth of two species in limited resource environment” 2014.Shangdong University, MA thesis.

7. Yano, Y et al. “Application of logistic growth model to pharmacodynamic analysis of in vitro bactericidal kinetics.” Journal of pharmaceutical sciences vol. 87,10 (1998): 1177-83. doi:10.1021/js9801337

Reaction parameters