Recent improvements in high-throughput biotechnologies such as microarray [1] and mass spectrometry [2] have led to various omics data showing gene expression, protein synthesis, metabolome flux, and cell-cell interactions [3] [4] [5] . The ensuing accumulation of omics data has contributed significantly to mathematical models that indicate dynamic characteristics of biological systems, including interactions between genes, proteins, cells, and tissues [6] [7] (Figure 1). Systems biology approaches such as mathematical modeling of multiple layers have revealed complex relationships among biological phenomena of varying spatiotemporal scales, and have elucidated mechanisms with high order functions in biological systems [8] [9] [10] [11] [12] . In particular, multi-time-scale models have been applied to analyses of intracellular signal transduction systems such as cell cycle control, cell fate determination, and immune system mechanisms [13] [14] [15] [16] . Moreover, mathematical analyses of varying (layer) gene (seconds), metabolism (minutes), and cell (hours) transition rates in biological systems define differences between biological systems and offer important discoveries of disease mechanisms. However, efficient techniques for numerical calculations remain elusive in practical applications of multi-time-scale mathematical models.

Multi-time-scale models comprise multiple layers that differ in rates of state change. During conventional numerical calculations of multi-time-scale models, time step sizes that are suitable for the smallest time-scale layer have been adopted for all layers to ensure stability and maintain calculation accuracy. Thus, dynamic behaviors of entire layers are numerically analyzed using excessively reduced time step sizes, leading to

significant increases in computational demands (time stiff problem) [17] [18] [19] . Numerous implicit methods such as the Radau method [19] and Gear method [20] have been proposed as candidate solutions to the time stiff problem. These methods generate numerical solutions based on calculation sensitivities and stabilities of components in 1 layer. Furthermore, the numerical solutions of these methods are calculated using non-linear simultaneous equations with n unknowns based on n components in the model. In calculating multi-time-scale model using the implicit method, larger time step sizes than those for the smallest time-scale layer can be applied to numerical calculations of all layers because the calculation stability of the implicit method is very high. However, because multi-time-scale models comprise large numbers of components, non-linear simultaneous equations that are calculated using implicit methods become very large. Specifically, although implicit methods suppress increases in computational loads due to excessive reductions in adaptive step sizes, significant increases in volumes of numerical calculations for non-linear simultaneous equations cause failure to eliminate the time stiff problem. Parallel computing with reduced computational cost has been applied to numerical calculations of multi-time-scale models [21] [22] [23] . In contrast, contributions of parallel computing have been limited because analyses of dynamic behaviors of biological systems include numerous sequential calculations. These observations imply that the efficiency of numerical calculations in multi-time-scale models is highly dependent on reduced computational loads. Therefore, application of suitable step sizes to numerical calculations for each time scale layer will likely reduce computational loads significantly. Currently, few methods are available for determining suitable step sizes for numerical calculations of each layer in multi-time-scale models with interactions among layers, and solutions to this problem are essential for practical applications of multi-time-scale models to biological systems.

In this study, we developed a method for dynamically determining appropriate step sizes for the largest time-scale layer based on state changes of the smallest time-scale layer in numerical analysis of multi-time-scale models with interactions among layers. Subsequently, we proposed a numerical method for reducing computational loads of multi-time-scale models (proposed method) and verified the effectiveness of the proposed method using the follow steps:

1) Construction of multi-time-scale model (benchmark model) with interactions among layers that are universally observed in biological systems;

2) Numerical calculation of benchmark models using the conventional method (Control);

3) Numerical calculation of a benchmark model using the proposed method;

4) Comparison of computational loads for proposed and conventional methods;

5) Comparison of numerical solutions for proposed and conventional methods;

6) Discussion of the validity of the proposed method.

Using these procedures, we demonstrated the utility of the proposed method for improving computational efficiency without increasing computational costs of multi- time-scale models with interactions among layers. By reducing computational loads, the proposed method enhances the feasibility of mathematical analyses and accommodates greater scales of mathematical models, representing a significant contribution to systems biology methods.