stochastic modeling
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1
Stochastic modeling
Visualize master equation Walk through stochastic simulation script
Ensemble distributions Exact particular trajectories
2
Master equations: Dynamics of population fractions
m (copies mRNA)
0 copies
0 DtTime
3
Master equations: Dynamics of population fractions
0 copies
Dt 2Dt0
m (copies mRNA)
4
Master equations: Dynamics of population fractions
0 copies
Dt 2Dt0 t t + Dt
− [ 𝑅↑ (𝑚 , 𝑡 )+𝑅↓ (𝑚 ,𝑡 ) ]∆ 𝑡 𝑁 (𝑚 ,𝑡 )+𝑅↑ (𝑚−1 ,𝑡 )∆ 𝑡 𝑁 (𝑚−1 ,𝑡 )+𝑅↓ (𝑚+1 ,𝑡 )∆ 𝑡 𝑁 (𝑚+1 ,𝑡 )𝑁 (𝑚 , 𝑡+∆ 𝑡 )=𝑁 (𝑚 ,𝑡 )𝑁 (𝑚 , 𝑡+∆ 𝑡 )−𝑁 (𝑚 , 𝑡 )=¿∆𝑁 (𝑚 ,𝑡 )=¿
+𝒪 (∆ 𝑡 2 )
m
m - 1
m + 1
5
Master equations: Dynamics of population fractions
0 copies
Dt 2Dt0 t t + Dt
∆𝑁 (𝑚 , 𝑡 )∆ 𝑡 𝑁 𝑇𝑂𝑇
=− [𝑅↑ (𝑚 ,𝑡 )+𝑅↓ (𝑚 ,𝑡 ) ] 𝑁 (𝑚 , 𝑡 )𝑁 𝑇𝑂𝑇
+𝑅↑ (𝑚−1 ,𝑡 ) 𝑁 (𝑚−1 , 𝑡 )𝑁 𝑇𝑂𝑇
+𝑅↓ (𝑚+1 ,𝑡 ) 𝑁 (𝑚+1 , 𝑡 )𝑁𝑇𝑂𝑇
− [ 𝑅↑ (𝑚 , 𝑡 )+𝑅↓ (𝑚 ,𝑡 ) ]∆ 𝑡 𝑁 (𝑚 ,𝑡 )+𝑅↑ (𝑚−1 ,𝑡 )∆ 𝑡 𝑁 (𝑚−1 ,𝑡 )+𝑅↓ (𝑚+1 ,𝑡 )∆ 𝑡 𝑁 (𝑚+1 ,𝑡 )∆𝑁 (𝑚 ,𝑡 )=¿
+𝒪 (∆ 𝑡 )
+𝒪 (∆ 𝑡 2 )
m
m - 1
m + 1
6
Master equations: Dynamics of population fractions
0 copies
Dt 2Dt0 t t + Dt
+𝒪 (∆ 𝑡 )
∆𝑃 (𝑚 ,𝑡 )∆ 𝑡 =− [𝑅↑ (𝑚 , 𝑡 )+𝑅↓ (𝑚 , 𝑡 ) ]𝑃 (𝑚 ,𝑡 )+𝑅↑ (𝑚−1 ,𝑡 )𝑃 (𝑚−1 ,𝑡 )+𝑅↓ (𝑚+1 , 𝑡 ) 𝑃 (𝑚+1 , 𝑡 )
∆𝑁 (𝑚 , 𝑡 )∆ 𝑡 𝑁 𝑇𝑂𝑇
=− [𝑅↑ (𝑚 ,𝑡 )+𝑅↓ (𝑚 ,𝑡 ) ] 𝑁 (𝑚 , 𝑡 )𝑁 𝑇𝑂𝑇
+𝑅↑ (𝑚−1 ,𝑡 ) 𝑁 (𝑚−1 , 𝑡 )𝑁 𝑇𝑂𝑇
+𝑅↓ (𝑚+1 ,𝑡 ) 𝑁 (𝑚+1 , 𝑡 )𝑁𝑇𝑂𝑇
+𝒪 (∆ 𝑡 )𝑑𝑃 (𝑚 , 𝑡 )
𝑑𝑡
m
m - 1
m + 1
Ensemble distributions Exact particular trajectories
7
Stochastic modeling
Visualize master equation Walk through stochastic simulation script
8
Walk-through of stochastic simulation script
1. Specify system chemistry2. Use current state vector to calculate time to next event3. Use current state vector to choose type of next event
9
Walk-through of stochastic simulation script
1. Specify system chemistry2. Use current state vector to calculate time to next event3. Use current state vector to choose type of next event
10
Specifying system
Type of event Rate Parameters Change to mRNA #
Change to protein #
Transcription kr kr +1 0
mRNA degradation
grx(1) gr -1 0
Protein synthesis kpx(1) kp 0 +1
Protein degradation
gpx(2) gp 0 -1
𝑥=[𝑥 (1 ) ¿ copies of mRNA𝑥 (2 ) ¿ copies of protein ]System variables:
System processes:
11
Specifying system
Type of event Rate Parameters Change to mRNA #
Change to protein #
Transcription kr kr +1 0
mRNA degradation
grx(1) gr -1 0
Protein synthesis kpx(1) kp 0 +1
Protein degradation
gpx(2) gp 0 -1
System variables:
System processes:
𝑥=[𝑥 (1 ) ¿ copies of mRNA𝑥 (2 ) ¿ copies of protein ]
12
Specifying system
Type of event Rate Parameters Change to mRNA #
Change to protein #
Transcription kr kr +1 0
mRNA degradation
grx(1) gr -1 0
Protein synthesis kpx(1) kp 0 +1
Protein degradation
gpx(2) gp 0 -1
System variables:
System processes:
𝑥=[𝑥 (1 ) ¿ copies of mRNA𝑥 (2 ) ¿ copies of protein ]
13
Specifying system
Type of event Rate Parameters Change to mRNA #
Change to protein #
Transcription kr kr +1 0
mRNA degradation
grx(1) gr -1 0
Protein synthesis kpx(1) kp 0 +1
Protein degradation
gpx(2) gp 0 -1
System variables:
System processes:
𝑥=[𝑥 (1 ) ¿ copies of mRNA𝑥 (2 ) ¿ copies of protein ]
14
Walk-through of stochastic simulation script
1. Specify system chemistry2. Use current state vector to calculate time to next event3. Use current state vector to choose type of next event
15
Calculate average firing rates for each independent channel
Type of event Rate Parameters Change to mRNA #
Change to protein #
Transcription kr kr +1 0
mRNA degradation
grx(1) gr -1 0
Protein synthesis kpx(1) kp 0 +1
Protein degradation
gpx(2) gp 0 -1
a1
a2
a3
a4
System variables:
System processes:
𝑥=[𝑥 (1 ) ¿ copies of mRNA𝑥 (2 ) ¿ copies of protein ]
Adding time-rates for individual channels
16
a1
a2
a3
a4
a0
Reaction channel firings in a population
17
a0
a0
a0
a0
a0
∆ 𝑡 ∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
a0
a0
a0
a0
a0
∆ 𝑡
18
Reaction channel firings in a population
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
a0
a0
a0
a0
a0
∆ 𝑡
19
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
Reaction channel firings in a population
20
a0
a0
a0
a0
a0
Reaction channel firings in a population
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡∆ 𝑡 1 𝑋∆ 𝑡 2 𝑋
21
a0
a0
a0
a0
a0
Reaction channel firings in a population
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡∆ 𝑡 1 𝑋∆ 𝑡 2 𝑋
22
a0
a0
a0
a0
a0
∆ 𝑡𝐿𝑂𝑁𝐺∆ 𝑡𝐿𝑂𝑁𝐺𝐸𝑅
Reaction channel firings in a population
a0
a0
a0
a0
23
a0
a0
a0
a0
a0
∆ 𝑡𝐕𝐄𝐑𝐘 𝑆𝐻𝑂𝑅𝑇
∆ 𝑡 𝐴𝐿𝑆𝑂 𝐕𝐄𝐑𝐘𝑆𝐻𝑂𝑅𝑇
Reaction channel firings in a populationZoomed in time scale
Dice represent shorter time intervals than before
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
Reaction channel firings in a population
24
a0
a0
a0
a0
a0
∆ 𝑡𝑆𝐻𝑂𝑅𝑇 ∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡∆ 𝑡𝐿𝑂𝑁𝐺
25
Draw duration from exponential distribution
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
26
Draw duration from exponential distribution
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
27
Draw duration from exponential distribution
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
28
Draw duration from exponential distribution
∆ 𝑃 𝐴𝑁𝑌 𝐸𝑉𝐸𝑁𝑇 ≅𝑎0∆ 𝑡
29
Draw duration from exponential distribution
30
Draw duration from exponential distribution
31
Draw duration from exponential distribution
32
Draw duration from exponential distribution
33
Draw duration until next event from exponential distribution
𝜏=𝑡0 ln( 1𝑅1 )
𝑅1=¿ 0 1
𝑡 0≔ ⟨𝜏 ⟩
34
Draw duration from exponential distribution
𝜏=𝑡0 ln( 1𝑅1 )
𝑅1=¿ 0 1
𝑡 0≔ ⟨𝜏 ⟩
𝜏=𝑡0 ln( 1𝑅1 )
𝑅1=¿ 0 1
35
𝜏=1.4 𝑡0
Draw duration from exponential distribution
¿1.4 /𝑎0
36
Walk-through of stochastic simulation script
1. Specify system chemistry2. Use current state vector to calculate time to next event3. Use current state vector to choose type of next event
37
Choose which event to perform
a1
a2
a3
a4
0
1
Type of event Rate Parameters Change to mRNA #
Change to protein #
Transcription kr kr +1 0
mRNA degradation
grx(1) gr -1 0
Protein synthesis kpx(1) kp 0 +1
Protein degradation
gpx(2) gp 0 -1
38
Walk-through of stochastic simulation script
1. Specify system chemistry2. Use current state vector to calculate time to next event3. Use current state vector to choose type of next event
39
Walk-through of stochastic simulation script
1. Specify system chemistry2. Use current state vector to calculate time to next event3. Use current state vector to choose type of next event
t
mRNA
Protein
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