TITLE:
An Effective Numerical Calculation Method for Multi-Time-Scale Mathematical Models in Systems Biology
AUTHORS:
Yohei Motomura, Hiroyuki Hamada, Masahiro Okamoto
KEYWORDS:
Finite Difference Method, Stiff Equation, Multi-Time-Scale, Systems Biology, Mathematical Analysis
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.17,
November
23,
2016
ABSTRACT: The improvements of high-throughput
experimental devices such as microarray and mass spectrometry have allowed an
effective acquisition of biological comprehensive data which include genome,
transcriptome, proteome, and metabolome (multi-layered omics data). In Systems
Biology, we try to elucidate various dynamical characteristics of biological
functions with applying the omics data to detailed mathematical model based on
the central dogma. However, such mathematical models possess multi-time-scale
properties which are often accompanied by time-scale differences seen among
biological layers. The differences cause time stiff problem, and have a grave
influence on numerical calculation stability. In the present conventional
method, the time stiff problem remained because the calculation of all layers
was implemented by adaptive time step sizes of the smallest time-scale layer to
ensure stability and maintain calculation accuracy. In this paper, we designed
and developed an effective numerical calculation method to improve the time
stiff problem. This method consisted of ahead, backward, and cumulative
algorithms. Both ahead and cumulative algorithms enhanced calculation
efficiency of numerical calculations via adjustments of step sizes of each
layer, and reduced the number of numerical calculations required for
multi-time-scale models with the time stiff problem. Backward algorithm ensured
calculation accuracy in the multi-time-scale models. In case studies which were
focused on three layers system with 60 times difference in time-scale order in
between layers, a proposed method had almost the same calculation accuracy
compared with the conventional method in spite of a reduction of the total
amount of the number of numerical calculations. Accordingly, the proposed
method is useful in a numerical analysis of multi-time-scale models with time
stiff problem.