Personalized large language models (LLMs) have attracted great attention in many applications, such as emotional support and role-playing. However, existing works primarily focus on modeling explicit character profiles, while ignoring the underlying personality traits that truly shape behaviors and decision-making, hampering the development of more anthropomorphic and psychologically-grounded AI systems. In this paper, we explore the modeling of Big Five personality traits, which is the most widely used trait theory in psychology, and propose P-React, a mixture of experts (MoE)-based personalized LLM. Particularly, we integrate a Personality Specialization Loss (PSL) to better capture individual trait expressions, providing a more nuanced and psychologically grounded personality simulacrum. To facilitate research in this field, we curate OCEAN-Chat, a high-quality, human-verified dataset designed to train LLMs in expressing personality traits across diverse topics. Extensive experiments demonstrate the effectiveness of P-React in maintaining consistent and real personality.