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Cell Cycle Control at the First Cell Cycle Control at the First Restriction Point and its Effect on Restriction Point and its Effect on
Tissue GrowthTissue Growth
Joint work with Avner Friedman and Bei Hu
(MBI, The Ohio State University and Notre Dame University)
Chiu-Yen Kao 高 秋 燕
2
Outline of the Talk
1. What is cell cycle?
2. Two restriction points (check points).
3. Mathematical Model: System of PDEs
4. Theorems
5. Numerical Demonstrations
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Cell Cycle
http://herb4cancer.files.wordpress.com/2007/11/cell-cycle2.jpg
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Cell Cycle
senegrosserppus
)(: duplicatedischromosomereplicatedisDNAS
MitosisGap
Gap
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Mathematical Model: Equations
1p2p
3p
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Mathematical Model: Boundary Condition
1
3p
1p 2p
2)(t
0p
4p
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Control at the First Restriction Point
1
3p
1p 2p
2)(t
0p
4p
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Conservation of Total Density
1
3p
1p 2p
2
4p
)(t0p
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Radial Symmetric VelocityOxygen Concentration
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Boundary Conditions & Initial Conditions
The global existence and uniqueness for radially symmetric solutions
is established by A. Friedman
Remark: We assume the oxygen level is above the necrotic level and growth rate and death rate are independent of oxygen concentration.
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Check point Control : constant case
)(t
1
3p
1p 2p
2)(t
0p
4p
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Check point Control : constant case
)(t
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Check point Control : constant case
)(t
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Check point Control : constant case
)(t
Idea: obtain a delay equation for Q
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Check point Control : constant case
)(t
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Check point Control : constant case
)(t
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Check point Control : constant case
)(t
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Check point Control : constant case
)(t
Idea: lower bound & upper bound
0 0
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Check point Control : constant case
)(t
Similarly,
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Numerical Simulation: Constant
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Numerical Simulation: Constant
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as Free Control)(t
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as Free Control)(t
Idea: Mathematical Induction Thus
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Numerical Simulation: as Free Control
)(t
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The case of variable )(t
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The case of variable )(t
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Conclusion
A multiscale model with two time scales: the usual time,t, and the running time of cells in each phase of the cell cycle.
The growth or shrinkage of a tissue depends on a decision that individual cells make whether to proceed directly from the first restriction point to the S phase or whether to go first into quiescent state.
The radius can be controlled by . With suitable , the radius is bounded above and below for all time.
The model can be easily extended to two (or more) populations of cells. (ex. healthy cell and tumor cell)
)(t )(t
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Selected ReferencesSelected References
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The EndThe End
