Biofilms are present in all natural, medical and industrial surroundings where bacteria live. Biofilm formation is a key factor in the growth and transport of both beneficial and harmful bacteria. While much is known about the later stages of biofilm formation, less is known about its initiation which is an important first step in the biofilm formation. In this paper, we develop a non-linear system of partial differential equations of Keller-Segel type model in one-dimensional space, which couples the dynamics of bacterial movement to that of the sensing molecules. In this case, bacteria perform a biased random walk towards the sensing molecules. We derive the boundary conditions of the adhesion of bacteria to a surface using zero-Dirichlet boundary conditions, while the equation describing sensing molecules at the interface needed particular conditions to be set. The numerical results show the profile of bacteria within the space and the time evolution of the density within the free-space and on the surface. Testing different parameter values indicate that significant amount of sensing molecules present on the surface leads to a faster bacterial movement toward the surface which is the first step of biofilm initiation. Our work gives rise to results that agree with the biological description of the early stages of biofilm formation.