Effectively engaging the public with science is vital for fostering trust and understanding in our scientific community. Yet, with an ever-growing volume of information, science communicators struggle to anticipate how audiences will perceive and interact with scientific news. In this paper, we introduce a computational framework that models public perception across twelve dimensions, such as newsworthiness, importance, and surprisingness. Using this framework, we create a large-scale science news perception dataset with 10,489 annotations from 2,101 participants from diverse US and UK populations, providing valuable insights into public responses to scientific information across domains. We further develop NLP models that predict public perception scores with a strong performance. Leveraging the dataset and model, we examine public perception of science from two perspectives: (1) Perception as an outcome: What factors affect the public perception of scientific information? (2) Perception as a predictor: Can we use the estimated perceptions to predict public engagement with science? We find that individuals' frequency of science news consumption is the driver of perception, whereas demographic factors exert minimal influence. More importantly, through a large-scale analysis and carefully designed natural experiment on Reddit, we demonstrate that the estimated public perception of scientific information has direct connections with the final engagement pattern. Posts with more positive perception scores receive significantly more comments and upvotes, which is consistent across different scientific information and for the same science, but are framed differently. Overall, this research underscores the importance of nuanced perception modeling in science communication, offering new pathways to predict public interest and engagement with scientific content.