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