Mass action

1

Law of mass action Cooperativity

Hill functions Bistability

fB

X

𝑑 𝑋𝑑𝑡 |

?

X

2

Law of mass action: A little lie . . .

𝑃 𝐿𝐸𝐹𝑇=𝑘𝑌𝑌

𝑃𝑇𝑂𝑃=𝑘𝑌 𝑌

𝑃 𝑅𝐼𝐺𝐻𝑇=𝑘𝑌𝑌

𝑃 𝐵𝑂𝑇𝑇𝑂𝑀=𝑘𝐵𝐵

Law of mass action: A little lie . . .

3

Law of mass action: A little lie . . .

4

𝑃 𝐿𝐸𝐹𝑇=𝑘𝑌𝑌

𝑃𝑇𝑂𝑃=𝑘𝑌 𝑌

𝑃 𝑅𝐼𝐺𝐻𝑇=𝑘𝑌𝑌

𝑃 𝐵𝑂𝑇𝑇𝑂𝑀=𝑘𝐵𝐵

Law of mass action: A little lie . . .

5

𝑑𝑅𝑑𝑡

=𝑘𝐷𝑘𝑌3 𝑘𝐵𝑀𝑌 3𝐵

Law of mass action: A little lie . . .

6

∆ 𝑡Between times and

𝑡 0

𝑑𝑅𝑑𝑡

=𝑘𝐷𝑘𝑌3 𝑘𝐵𝑀𝑌 3𝐵

Law of mass action: A little lie . . .

7

𝑑𝑅𝑑𝑡

=𝑘𝑂𝑉𝐸𝑅𝐴𝐿𝐿𝑋𝜈𝑥𝑌𝜈 𝑦 𝑍𝜈𝑧⋯

X

Y

ZZ

Z

Y

Y

X

X

XX

XYZ

3𝑌 +1𝑀+1𝐵𝑘𝐷𝑘𝑌3 𝑘𝐵

→

𝑅

𝜈𝑥 𝑋+𝜈𝑦𝑌 +𝜈𝑧𝑍+⋯𝑘𝑂𝑉𝐸𝑅𝐴𝐿𝐿

→

𝑅

Mass action

8

Law of mass action Cooperativity

Hill functions Bistability

fB𝑑 𝑋𝑑𝑡 |

?

X X

𝑑𝐸𝐵

𝑑𝑡 |𝐵𝐼𝑁𝐷

=𝜕𝐸𝐵

𝜕𝑅𝐵

𝑑𝑅𝐵

𝑑𝑡

𝑑𝐸𝐵

𝑑𝑡 |𝑈𝑁𝐵𝐼𝑁𝐷

=𝜕𝐸𝐵

𝜕𝑅𝑈

𝑑 𝑅𝑈

𝑑𝑡

0=𝑑 𝐸𝐵

𝑑𝑡=𝑑𝐸𝐵

𝑑𝑡 |𝐵𝐼𝑁𝐷

+𝑑𝐸𝐵

𝑑𝑡 |𝑈𝑁𝐵𝐼𝑁𝐷

=𝑘𝐵𝐼𝑁𝐷 𝑋4𝐸𝐹−𝑘𝑈𝑁𝐵𝐼𝑁𝐷 𝐸𝐵

𝑓 𝐵=𝑋 4

𝑋 4+𝑘𝑈𝑁𝐵𝐼𝑁𝐷

𝑘𝐵𝐼𝑁𝐷

XX

XX

9

XX

XX

XX

XX

XX

XX

𝑘𝐵𝐼𝑁𝐷𝑋4𝐸𝐹 𝑘𝑈𝑁𝐵𝐼𝑁𝐷 𝐸𝐵+1 -1

𝑘𝑈𝑁𝐵𝐼𝑁𝐷 𝐸𝐵=𝑘𝐵𝐼𝑁𝐷 𝑋4𝐸𝐹

𝐸𝐵

(𝐸𝑇𝑂𝑇−𝐸𝐵 )

(𝑘𝐵𝐼𝑁𝐷 𝑋4+𝑘𝑈𝑁𝐵𝐼𝑁𝐷 ) 𝐸𝑇𝑂𝑇𝑘𝐵𝐼𝑁𝐷𝑋

4¿

EF EB XX

XX

Cooperativity and Hill functions: Another lie . . .

𝑋 1/24

10

X

X

X

X

XX

XX

𝑓 𝐵=𝑋 4

𝑋 4+𝑋 1 /24  

Cooperativity and Hill functions: Another lie . . .

𝑓 𝐵=𝑋𝑛

𝑋𝑛+𝑋 1/2𝑛  X

X

X

X

X

X

. . .

. . .

n dimples n bound dimples

𝑓 𝐵=𝑋𝑛

𝑋𝑛+𝑋 1/2𝑛  X

X

XXX

X

11

Cooperativity and Hill functions: Another lie . . .

fB

X

XX

X1/2

1

0

𝑓 𝐵=𝑋𝑛

𝑋𝑛+𝑋 1/2𝑛  X

X

XXX

X

12

Cooperativity and Hill functions: Another lie . . .

fB

X

XX

XX

XX

𝑓 𝐵=𝑋𝑛

𝑋𝑛+𝑋 1/2𝑛  X

X

XXX

X

13

Cooperativity and Hill functions: Another lie . . .

fB

X

𝑓 𝐵=𝑋𝑛

𝑋𝑛+𝑋 1/2𝑛  X

X

XXX

X

14

Cooperativity and Hill functions: Another lie . . .

fB

X

n = 1

n = 3n = 2

n = 4

n = 8n = 16

X1/2

Mass action

15

Law of mass action Cooperativity

Hill functions Bistability

fB𝑑 𝑋𝑑𝑡 |

?

X X

𝑑 𝑋𝑑𝑡 |

𝐷𝐸𝐺𝑅

= 𝜕 𝑋𝜕 𝑅𝐷

𝑑𝑅𝐷

𝑑𝑡𝑑 𝑋𝑑𝑡 |

𝐶𝐴𝑇

= 𝜕 𝑋𝜕𝑅𝐶

𝑑𝑅𝐶

𝑑𝑡

Bistability

16

X

X

X

X

XX

XX W

X X

f

𝑓 𝐵=𝑋𝑛

𝑋𝑛+𝑋 1/2𝑛  

𝛽𝑊 𝐸𝑇𝑂𝑇 𝑓 𝐵+1𝑋𝑛

𝑋𝑛+𝑋1 /2𝑛  

~𝛽 -1 𝛼 𝑋

𝑑 𝑋𝑑𝑡

= 𝑑 𝑋𝑑𝑡 |

𝐶𝐴𝑇

+ 𝑑 𝑋𝑑𝑡 |

𝐷𝐸𝐺𝑅

=~𝛽 𝑋𝑛

𝑋𝑛+𝑋 1 /2𝑛  −𝛼 𝑋

𝑑 𝑋𝑑𝑡 |

?

X

~𝛽

0

Bistability

17

X

X

X

X

XX

XX W

X X

f

𝑑 𝑋𝑑𝑡

=~𝛽 𝑋𝑛

𝑋𝑛+𝑋 1 /2𝑛  −𝛼𝑋

X

𝑑 𝑋𝑑𝑡

𝑡

𝑋 (𝑡 )

0

0