###### Abstract  
The hydrolysis time is directly related to the flavor of the Maillard reaction, but existing proxy models cannot simulate and model the variation curves of vital volatile components. This study developed a predictive model for modelling and simulating key volatile compounds of Maillard reaction products (MRPs) derived from beef fallow residue hydrolysate. Results showed the degree of hydrolysis increased with hydrolysis time, and the most significant improvement in the roast flavor and overall acceptance was when hydrolyzing 4 h. Based on flavor dilution value and the relative odor activity value, nine key volatile components were identified, and 2-ethyl-3.5-dimethylpyrazine with roast flavor was the highest. Compared with Polynomial Curve Fitting (PCF) and Cubic Spline Interpolation (CSI), key volatile compounds of MRPs could be better modeled and simulated by the Curve Prediction Model (CPM). All results suggested that CPM could predict the changes in key volatile components produced by MRPs. hydrolysis time; Maillard reaction products; cubic spline interpolation; polynomial curve fitting; curve prediction model +
Footnote †: journal: _foods_  
0000-0001-9900-0001-9900-0001-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-9900-99000-99000-99000-9900-99000-99000-99000-99000-99  
compounds, including esters, acids, alcohols and aldehydes, was significantly improved by the hydrolysis of pork sarcoplasmic protein and myofibrillar protein [4]. However, the effect of hydrolysis on the volatile compounds of Maillard reaction products derived from beef tallow residue is still unknown. However, several factors can affect the process. For instance, some scholars found that hydrolysis was affected by many factors: hydrolysis time, protease dosage, hydrolysis temperature, pH, solid-to-liquid ratio, etc., that affect the hydrolysis of Litopenaeus vannamei [5]. This study concluded that the essential factor was hydrolysis time. According to the research of some scholars, soybean protein immunoreactivity increased and decreased with different hydrolysis times [6]. They found that controlled hydrolysis conditions combined with controlled Maillard induction did not significantly decrease the immune activity. Partial least squares regression (PLSR) has been commonly used to predict the Maillard reactions with different degrees of hydrolysis (DH) based on a point-to-line basis. Some studies carried out a correlation analysis between the molecular weight of peptides, odor-active compounds and sensory evaluation of beef with various DH by using PLSR and found that beef matrix with DH 29.13 was an ideal precursor to impart aroma characteristics of beef processing flavor [7]. Some researchers developed a PLSR model based on Fourier transform infrared spectroscopy to predict DH using the molecular weight of milk protein hydrolysate, indicating that PLSR is a promising tool for the prediction of DH in the Maillard reactions [8]. Although PLSR is precise and sensitive, the large amount of experimental data required makes it time-consuming. Recently, proxy models have become popular with the development of computation, including Cubic Spline Interpolation (CSI), Polynomial Curve Fitting (PCF) and Curve Prediction Model (CPM). Proxy models could resolve the sequence integrity and low-reliability issues without requiring a large sample space of data and also generate irregular raw data to construct an intense generated sequence to make it highly accurate and sensitive in dealing with a small amount of data. For example, CSI could handle curve simulation with sampled points and has ideal geometric characteristics, smooth curves, high fitting precision and practicability [9], and the effect of temperature on crop yields could be well predicted by CSI [10]. For PCF, it provides discontinuity and asymmetry in fitting curves, which are suitable for anatomical curves [11].