Steganography is an important technology for ensuring the security of cyberspace and the privacy of communications. In the last decade, emerging biotechnology has made it possible for DNA to be used as a promising steganographic carrier with high hidden capacity, high imperceptibility and high feasibility. However, severe statistical distortion might appear in steganographic carriers generated by existing DNA steganographies when they are compared with the natural ones. Therefore, efforts are being made to seek an advanced strategy to generate quasi-natural steganographic carriers with a strong anti-steganalysis capability. In this work, we first thoroughly analyze and model the numerous complicated statistical properties that exist in natural DNA chains, and then utilize the LSTM model to learn the serialized statistical properties. After obtaining an optimal sequence model that highly satisfies the statistical properties of natural DNA chains, we utilize the Adaptive Dynamic Grouping (ADG) algorithm to perform information hiding. In addition, we have carried out experimental analysis and verification from the perspectives of perceptual-imperceptibility, statistical-imperceptibility, and anti-steganalysis capability, all of which show that our proposed steganography method vastly outperforms previous DNA steganographic methods, taking a successful step towards achieving higher security DNA steganography.