Canonical protein dynamics

1

Differential equationFlowchart

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥f

𝛽 𝛼𝑥

𝛽𝛼

𝑥 (𝑡 )

𝑡0

Solution

2

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

No translations

Physical picture of protein translation and degradation

No degradation for this protein

+No

chan

ge

Physical picture of protein translation and degradation

3

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

Chance translation

+

+

Physical picture of protein translation and degradation

4

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

No

chan

ge

Neglecting to roll dice on this freshly

generated protein.

+

Physical picture of protein translation and degradation

5

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+N

o ch

ange

+

Physical picture of protein translation and degradation

6

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

Physical picture of protein translation and degradation

7

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

No

chan

ge

Rolled dice for protein that was

already degraded!

+

Physical picture of protein translation and degradation

8

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

Physical picture of protein translation and degradation

9

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

Physical picture of protein translation and degradation

10

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

Chance translation

+

Physical picture of protein translation and degradation

11

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

+

Physical picture of protein translation and degradation

12

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

+

Physical picture of protein translation and degradation

17

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

+

Physical picture of protein translation and degradation

18

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ + +

+

+

Chance translation

+

Physical picture of protein translation and degradation

19

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ + +

+

+

+

Physical picture of protein translation and degradation

20

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ + +

+

+

+

Physical picture of protein translation and degradation

21

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ + +

+

+

+

Physical picture of protein translation and degradation

22

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

+

∆ 𝑡

Between times and

≅2degradations

(20 time steps ) (28 proteins )(¿ time steps ) (¿ proteins )Proteins lost

≅3 translations20 time steps

(¿ time steps )Proteins added ≅ 𝛽∆ 𝑡

≅ 𝛼∆ 𝑡 𝑥 (𝑡 0 )𝑥 (𝑡 0+∆𝑡 )≅ 𝑥 (𝑡 0 )+𝛽∆ 𝑡−𝛼∆ 𝑡 𝑥 (𝑡 0 )

𝑥 (𝑡 0+∆𝑡 )−𝑥 (𝑡 0 )≅ 𝛽∆ 𝑡−𝛼∆ 𝑡 𝑥 (𝑡 0 )

𝑑𝑥𝑑𝑡 |𝑡=𝑡 0≅

𝑥 (𝑡0+∆ 𝑡 )−𝑥 (𝑡0 )∆ 𝑡

≅ 𝛽−𝛼 𝑥 ( 𝑡0 )

𝑡 0 +

Physical picture of protein translation and degradation

23

𝑡 (𝜏𝐷𝐼𝐶𝐸 )2 3 4 5 6 7 8 90 1 12 13 14 15 16 17 18 1910 11 20

Note: Proteins can be distinguished by momentary distortions.

+ +

+

+

+

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥

+

STOPLimitation!!! (But we all do it this it this way)

x (# proteins)

28

29

30

Realistic trajectory

Unrealistically smooth trajectory

f𝛽 𝛼

𝑥

Canonical protein dynamics

24

Differential equationFlowchart

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥f

𝛽 𝛼𝑥

𝛽𝛼

𝑥 (𝑡 )

𝑡0

Solution

𝛽𝛼

Time-course and rise time

25

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥f

𝛽 𝛼𝑥

Declare initial condition, for example, “start at time 0 with no protein”

𝑥 (0 )=0

Look for possible steady states

0=𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥𝑆𝑇

𝑥 (𝑡 )

𝑡0

𝑥𝑆𝑇=𝛽𝛼

Estimate some slopes

Time-course and rise time

26

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥f

𝛽 𝛼𝑥

𝑥 (𝑡 )

𝑡

0

𝛽𝛼

𝑑𝑥𝑑𝑡

=𝛼 ( 𝛽𝛼−𝑥 )𝑠 (𝑥 )

𝑠 (𝑥 )=−𝑥+𝛽𝛼

𝑠 (𝑥 (0 ) )= 𝛽𝛼

𝑠 (𝑥𝑆𝑇 )=− 𝛽𝛼 +𝛽𝛼

=0

𝑑 𝑠𝑑𝑡

=𝑑𝑠𝑑𝑥

𝑑𝑥𝑑 𝑡

𝑑 𝑠𝑑𝑡

= 𝑑𝑑𝑥 (− 𝑥+ 𝛽

𝛼 ) 𝑑𝑥𝑑𝑡

−𝑑𝑠𝑑 𝑡

=𝛼 𝑠

∫𝑡=0

𝑡 𝑓1𝑠𝑑 𝑠𝑑𝑡

𝑑𝑡=−𝛼 ∫𝑡=0

𝑡 𝑓

𝑑𝑡

∫𝑠=𝑠 (𝑥 (0 ))

𝑠 ( 𝑥 (𝑡 𝑓 ) )1𝑠𝑑 𝑠=−𝛼 ∫

𝑡=0

𝑡 𝑓

𝑑𝑡

𝑑 𝑠𝑑𝑡

=−𝑑𝑥𝑑𝑡

Time-course and rise time

27

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥f

𝛽 𝛼𝑥

𝑥 (𝑡 )

𝑡

0

𝛽𝛼

𝑠 (𝑥 )=−𝑥+𝛽𝛼

𝑠 (𝑥 (0 ) )= 𝛽𝛼

𝑠 (𝑥𝑆𝑇 )=− 𝛽𝛼 +𝛽𝛼

=0

𝑑 𝑠𝑑𝑡

=𝑑𝑠𝑑𝑥

𝑑𝑥𝑑 𝑡

𝑑 𝑠𝑑𝑡

= 𝑑𝑑𝑥 (− 𝑥+ 𝛽

𝛼 ) 𝑑𝑥𝑑𝑡

∫𝑠=𝑠 (𝑥 (0 ))

𝑠 ( 𝑥 (𝑡 𝑓 ) )1𝑠𝑑 𝑠=−𝛼 ∫

𝑡=0

𝑡 𝑓

𝑑𝑡

ln (𝑠 (𝑥 ( 𝑡 𝑓 )) )− ln( 𝛽𝛼 )=−𝛼 [𝑡 𝑓 −0 ]

𝑑 𝑠𝑑𝑡

=−𝑑𝑥𝑑𝑡

ln ( 𝑠 (𝑥 (𝑡 𝑓 ) )𝛽𝛼 )=−𝛼𝑡 𝑓𝑠 (𝑥 (𝑡 𝑓 ) )

𝛽𝛼

=𝑒−𝛼𝑡 𝑓

𝑠=𝛽𝛼𝑒−𝛼𝑡 𝑓

𝑇 12

Time-course and rise time

28

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥f

𝛽 𝛼𝑥

𝑥 (𝑡 )

𝑡0

𝑠=𝛽𝛼𝑒−𝛼𝑡 𝑓

𝛽2𝛼

𝛽𝛼

𝛽2𝛼

𝛽𝛼

𝛽2𝛼

𝛽𝛼

𝛽2𝛼

𝛽𝛼

𝛽2𝛼

𝛽𝛼

𝑇 12

=ln (2 )𝛼

STOPShow

𝑇 12

𝑇 12

𝑇 12

𝑇 12

𝑇 12

𝛽2𝛼

Time-course and rise time

29

𝑑𝑥𝑑𝑡

=𝛽−𝛼 𝑥f

𝛽 𝛼𝑥

𝑥 (𝑡 )

𝑡0

𝑠=𝛽𝛼𝑒−𝛼𝑡 𝑓

𝛽𝛼

𝑇 12

=ln (2 )𝛼