Modelling the influence of vitamin D and probiotic supplementation on the microbiome and immune response

The intestinal microbiota play a critical role in human health and disease, maintaining metabolic and immune/inflammatory health, synthesizing essential vitamins and amino acids and maintaining intestinal barrier integrity. The aim of this paper is to develop a mathematical model to describe the complex interactions between the microbiota, vitamin D/vitamin D receptor (VDR) pathway, epithelial barrier and immune response in order to understand better the effects of supplementation with probiotics and vitamin D. This is motivated by emerging data indicating the beneficial effects of vitamin D and probiotics individually and when combined. We propose a system of ordinary differential equations determining the time evolution of intestinal bacterial populations, concentration of the VDR:1,25(OH)$_{2}$D complex in epithelial and immune cells, the epithelial barrier and the immune response. The model shows that administration of probiotics and/or vitamin D upregulates the VDR complex, which enhances barrier function and protects against intestinal inflammation. The model also suggests co-supplementation to be superior to individual supplements. We explore the effects of inflammation on the populations of commensal and pathogenic bacteria and the vitamin D/VDR pathway and discuss the value of gathering additional experimental data motivated by the modelling insights.