Post on 06-Feb-2016
description
transcript
Canonical protein dynamics
1
Differential equationFlowchart
ππ₯ππ‘
=π½βπΌ π₯f
π½ πΌπ₯
π½πΌ
π₯ (π‘ )
π‘0
Solution
2
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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
π‘ (ππ·πΌπΆπΈ )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.
+ +
+
+
+
β π‘
Between times and
β 2degradations
(20 time steps ) (28 proteins )(ΒΏ time steps ) (ΒΏ proteins )Proteins lost
β 3 translations20 time steps
(ΒΏ time steps )Proteins added β π½β π‘
β πΌβ π‘ π₯ (π‘ 0 )π₯ (π‘ 0+βπ‘ )β π₯ (π‘ 0 )+π½β π‘βπΌβ π‘ π₯ (π‘ 0 )
π₯ (π‘ 0+βπ‘ )βπ₯ (π‘ 0 )β π½β π‘βπΌβ π‘ π₯ (π‘ 0 )
ππ₯ππ‘ |π‘=π‘ 0β
π₯ (π‘0+β π‘ )βπ₯ (π‘0 )β π‘
β π½βπΌ π₯ ( π‘0 )
π‘ 0 +
Physical picture of protein translation and degradation
23
π‘ (ππ·πΌπΆπΈ )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.
+ +
+
+
+
ππ₯ππ‘
=π½βπΌ π₯
+
STOPLimitation!!! (But we all do it this it this way)
x (# proteins)
28
29
30
Realistic trajectory
Unrealistically smooth trajectory
fπ½ πΌ
π₯
Canonical protein dynamics
24
Differential equationFlowchart
ππ₯ππ‘
=π½βπΌ π₯f
π½ πΌπ₯
π½πΌ
π₯ (π‘ )
π‘0
Solution
π½πΌ
Time-course and rise time
25
ππ₯ππ‘
=π½βπΌ π₯f
π½ πΌπ₯
Declare initial condition, for example, βstart at time 0 with no proteinβ
π₯ (0 )=0
Look for possible steady states
0=ππ₯ππ‘
=π½βπΌ π₯ππ
π₯ (π‘ )
π‘0
π₯ππ=π½πΌ
Estimate some slopes
Time-course and rise time
26
ππ₯ππ‘
=π½βπΌ π₯f
π½ πΌπ₯
π₯ (π‘ )
π‘
0
π½πΌ
ππ₯ππ‘
=πΌ ( π½πΌβπ₯ )π (π₯ )
π (π₯ )=βπ₯+π½πΌ
π (π₯ (0 ) )= π½πΌ
π (π₯ππ )=β π½πΌ +π½πΌ
=0
π π ππ‘
=ππ ππ₯
ππ₯π π‘
π π ππ‘
= πππ₯ (β π₯+ π½
πΌ ) ππ₯ππ‘
βππ π π‘
=πΌ π
β«π‘=0
π‘ π1π π π ππ‘
ππ‘=βπΌ β«π‘=0
π‘ π
ππ‘
β«π =π (π₯ (0 ))
π ( π₯ (π‘ π ) )1π π π =βπΌ β«
π‘=0
π‘ π
ππ‘
π π ππ‘
=βππ₯ππ‘
Time-course and rise time
27
ππ₯ππ‘
=π½βπΌ π₯f
π½ πΌπ₯
π₯ (π‘ )
π‘
0
π½πΌ
π (π₯ )=βπ₯+π½πΌ
π (π₯ (0 ) )= π½πΌ
π (π₯ππ )=β π½πΌ +π½πΌ
=0
π π ππ‘
=ππ ππ₯
ππ₯π π‘
π π ππ‘
= πππ₯ (β π₯+ π½
πΌ ) ππ₯ππ‘
β«π =π (π₯ (0 ))
π ( π₯ (π‘ π ) )1π π π =βπΌ β«
π‘=0
π‘ π
ππ‘
ln (π (π₯ ( π‘ π )) )β ln( π½πΌ )=βπΌ [π‘ π β0 ]
π π ππ‘
=βππ₯ππ‘
ln ( π (π₯ (π‘ π ) )π½πΌ )=βπΌπ‘ ππ (π₯ (π‘ π ) )
π½πΌ
=πβπΌπ‘ π
π =π½πΌπβπΌπ‘ π
π 12
Time-course and rise time
28
ππ₯ππ‘
=π½βπΌ π₯f
π½ πΌπ₯
π₯ (π‘ )
π‘0
π =π½πΌπβπΌπ‘ π
π½2πΌ
π½πΌ
π½2πΌ
π½πΌ
π½2πΌ
π½πΌ
π½2πΌ
π½πΌ
π½2πΌ
π½πΌ
π 12
=ln (2 )πΌ
STOPShow
π 12
π 12
π 12
π 12
π 12
π½2πΌ
Time-course and rise time
29
ππ₯ππ‘
=π½βπΌ π₯f
π½ πΌπ₯
π₯ (π‘ )
π‘0
π =π½πΌπβπΌπ‘ π
π½πΌ
π 12
=ln (2 )πΌ