## 1 Introduction  
Regulation of gene expression remains a pivotal subject of research, particularly in transgenic engineering [1]. Theoretical frameworks have extensively utilized one-dimensional sequence data from gene-transcribed regions to elucidate the interactions between regulatory elements and transgene expression levels [2, 3]. These studies aimed to enhance exogenous gene expression; however, challenges in eukaryotic transgene regulation persist, necessitating the development of controlled, transgenic expression systems. Enhanced promoter strength increases transgene expression [4]. However, owing to the intricate nature of eukaryotic gene regulation, not all robust promoters can achieve maximal protein expression and activity [5]. Increasing the gene copy number is a straightforward approach; however, excessive copy numbers can be detrimental [6]. Codon optimization within protein-coding sequences is a common strategy for boosting gene expression [7, 8, 9]. Synonymous codon usage frequencies vary significantly among species such as _Escherichia coli_[10], yeasts [11], animals [12], and plants [13]. Synonymous codon substitutions can alter post-translational modification sites, thereby affecting the protein structure, stability, and function [14, 15]. Furthermore, enhancers can improve transgeneexpression but must be paired with strong promoters and may inadvertently affect host gene expression [3; 16]. Eukaryotic chromosomes are organized into nucleosome arrays, with nucleosomes comprising \(\sim\)147 bp DNA segments wrapped around histone octamers, each containing two copies of histones H2A, H2B, H3, and H4 [17; 18]. Nucleosome positioning modulates the binding of DNA-binding proteins, thereby influencing gene transcription and expression [19; 20]. In the transcriptional start region, there are two relatively well-positioned nucleosomes located upstream and downstream of the transcription start sites (TSSs), which are referred to as the \(-1\) nucleosome and the \(+1\) nucleosome, respectively [21; 22]. Between the \(-1\) nucleosome and the \(+1\) nucleosome, there is a variable-length nucleosome-depleted region (NDR) [23]. NDRs are prevalent across transcriptional regions in various organisms including yeast [24], plants [25], _D. melanogaster_[26], _C. elegans_[27], and humans [28; 29]. Positioning of the \(-1\) nucleosome upstream of the NDR is highly variable, whereas the \(+1\) nucleosome downstream of the NDR remains relatively stable [30]. NDR serves as a critical functional element for transcription initiation, is enriched with TFBSs, and facilitates transcription factor entry into the chromatin. Transcription initiation is triggered by the binding of transcription factors, and the efficiency of this binding directly influences the transcription efficiency [31; 32; 33]. In _Saccharomyces cerevisiae_, the TSSs of most genes are located at the \(3^{\prime}\)-end of NDRs or within the \(+1\) nucleosome. The \(-1\) and \(+1\) nucleosomes, along with the intervening NDR sequences, play pivotal roles in the regulation of gene expression [34; 35; 36]. The efficiency of transcription factor (TF) binding to NDRs is closely linked to NDR length [37; 38; 39]. Generally, TFs bind to NDRs in three modes: (1) When the NDR is sufficiently long, TFs can easily bind without requiring upstream movement of the \(-1\) nucleosome, resulting in high binding efficiency and typically high gene expression. (2) With shorter NDRs, the space for TF binding is limited, necessitating upstream movement of the \(-1\) nucleosome to extend the NDR sequence. Nucleosome repositioning consumes energy, reduces the binding efficiency, and leads to moderate or weak gene expression. (3) For very short NDRs, a TF-binding space can be created by displacing the \(-1\) nucleosomes. Thus, the NDR length between the \(-1\) and \(+1\) nucleosomes dictates TF binding efficiency and transcription efficacy. Regulating transgenic expression based solely on one-dimensional DNA information has limitations.