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Competition model

Purpose

1. To quantize and prove the saying of eliminating all S. mutans would lead to the rise of other bacteria via using mathematical model.
2. To predict the change of pH and ecology in oral cavity after eliminating different amount of S. mutans.

Background

Dental caries is defined as demineralization of tooth enamel[1]. Because tooth enamel solubility is pH dependent[2], acid production in plaque area is believed to be crucial for caries development. In 1940, Stephan has discovered that the longer time pH value of oral cavity is below “critical pH 5.5”, the more susceptible tooth enamel is. Therefore, NYMU team plans to prevent tooth caries via eliminating S. mutans, the main cariogenic bacteria that produce most of the acid[3].

However, studies have shown that when S. mutans no longer adapts to the environment, or the amount of S. mutans decreased to an extent, other species will become dominant instead[4]. Thus, there is concern that if we kill S. mutans excessively, other bacteria population would grow, and cause caries or do other harm to the oral cavity. Due to the limitation of our lab, we cannot do experiment in vitro or in multi-species culture to verify the saying. We then use modelling, the competition model, to demonstrate the reason why we cannot kill all S. mutans, and find the optimize amount of S. mutans elimination to prevent caries without concerning other cariogenic bacteria.

In our competition model, we choose to use Lotka–Volterra competition model, which based on logistic equation, for it fits very well with our experimental data on S. mutans growth curve. We choose four species that can produce acid and occupy high proportion in our oral cavity as the subjects of modeling, which are S. mutans, S. sobrinus, S. mitis[5], and Neisseria mucosa[6]. To validate our model more realistically and precisely, we use 16S rRNA gene sequencing data from a paper published in 2012, which samples from 36 human[7], to find out how bacteria compete in oral cavity. Moreover, we use experimental data from literature to know how populations shift effect oral pH value. Then we can find out the optimal S. mutans population that would maximize oral pH value, and therefore prevent caries from happening.

Models and mathematic equations

Competitive Lotka–Volterra equation presents the competition between two or more species for limiting resources[8].

$$\frac{dx_i}{dt}=r_i x_i (1-\frac{\sum_{j=1}^N a_{ij} x_j}{K_i})$$

The model is based on logistic growth equation.

$$\frac{dx}{dt}=r x (1-\frac{x}{K})$$

The $(1-\frac{x}{K})$ term means growth rate is limited by intraspecific competition, while x is population size and K is the maximum population size[9]. The matrix a$(\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1N}\\a_{21} & a_{22} & \cdots & a_{2N}\\ \vdots & \vdots & \ddots & \vdots\\a_{N1} & a_{N2} & \cdots & a_{NN} \end{bmatrix})$ represent the interspecies interaction. $a_{12}$ represents the effect species 2 has on the population of species 1. For example, if species 1 is a strong competitor, $a_{21}, a_{31}, a_{41}$...would have greater value, which turns out to negatively influence other species by reducing population sizes.

We choose Lotka–Volterra competition model because it demonstrate multispecies competition clearly, and its basis, logistic growth curve, fits well with our experimental data(see growth & pH model).

Result and model validation

Reference