Abstract
The findings in this study suggest that the solution of a boundary value problem for differential equation system can be used to discuss the fencing problem in mathematics and Coleochaete, a green algae, cell division. This differential equation model in parametric expression is used to simulate the two kinds of cell division process, one is for the usual case and the case with a “dead” daughter cell.
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Acknowledgments
The authors would like to thank the Editor/referee for the valuable comments on our submitted manuscript. And we also would like to thank Prof. Heping Ma and Dr. Hamdi Zorgati for their suggestion. The author Zhigang Zhou thanks Dr. Haseloff in the University of Cambridge for helping to get these Coleochaete images.
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The research was supported partly by a grant of The First-Class Discipline of Universities in Shanghai.
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Appendix
Appendix
1.1 Experimental data
The picture Fig. 2 is only a selection of all experiment records of 280 h. The radial division and the tangent division in the experiment can be showed in the following Table 1. The exception is with “\(\dag \)”. The notation of \(\theta ^*\) is the theoretic value coming from (3.1), and \(\theta \) is the measurement in the experiment. All of natations has the meaning as in Fig. 6.
1.2 Uniform cell division
The notation of \(\theta _1\) in Table 2 is the angle between \(x\)-axis and the beginning side \(\theta _2\) is between \(x\)-axis and the ending side. And \(\theta _3=(\theta _2-\theta _1)/2\) is the angle for radial dividing line. We denote by \(r\) the radius of the disc, and by “Center” and \(R\) the center and the radius of dividing arc respectively. The letter of “R” denotes the radial division and “T” is for tangent.
1.3 Non-uniform cell division
The notation \(r\) in Table 3 and Table 4 is the center of the arc in the first quadrant, “Center-1” and \(r_1\) are the center and the radius of the southeast to the first quadrant respectively. “Center-2” and \(r_2\) are the center and the radius of dividing arc respectively. The radius \(R\) of piece of circle in the second quadrant is equal to \(r+1.04812.\, \theta _1\) and \(\theta _2\) are the angles of the beginning side and the ending side for the sector in the first quadrant respectively. The sector’s center is at the point \((0,1.04812)\). The “Arched” is the half circle in the southeast to the first quadrant. \(\theta _3\) in Table 4 is the angle of the dividing line in the sector.
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Wang, Y., Dou, M. & Zhou, Z. The fencing problem and Coleochaete cell division. J. Math. Biol. 70, 893–912 (2015). https://doi.org/10.1007/s00285-014-0784-9
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DOI: https://doi.org/10.1007/s00285-014-0784-9