Molecular communication (MC) enables conveying information at nano- to micro-scales via molecules or nanoscale particles. MC systems for fixed transmitter and receiver nanomachines have been extensively investigated. However, the scenarios for mobile MC are seldomly studied. In the paper, the mutual information and achievable rate in the presence of inter-symbol interference and noise for the mobile MC is investigated. The scenario that both the transmitter and the receiver are in independent random motions is considered. Due to the mobility, the channel impulse response varies and results in a varying received signal with varying signal-dependent noise. This further leads to the random variation of the probability density functions of received concentration for symbols which further generates varying error probability and therefore varying mutual information and maximum achievable rate. In this paper, the mutual information and maximum achievable rate for mobile MC scenario are derived and analyzed. The impacts of different parameters, such as the initial transmitter-receiver distance, the number of released molecules, symbol interval, the mobility of nanomachine, and priori probabilities of the transmitted symbol on the mutual information and maximum achievable rate in mobile MC scenario are studied by simulation investigations.