stochastic modeling
DESCRIPTION
Stochastic modeling. Walk through stochastic simulation script. Visualize master equation. Exact particular trajectories. Ensemble distributions. Master equations: Dynamics of population fractions. m (copies mRNA). 0 copies. D t. Time. 0. - PowerPoint PPT PresentationTRANSCRIPT
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Stochastic modeling
Visualize master equation Walk through stochastic simulation script
Ensemble distributions Exact particular trajectories
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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)
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Master equations: Dynamics of population fractions
0 copies
Dt 2Dt0 t t + Dt
β [ π β (π , π‘ )+π β (π ,π‘ ) ]β π‘ π (π ,π‘ )+π β (πβ1 ,π‘ )β π‘ π (πβ1 ,π‘ )+π β (π+1 ,π‘ )β π‘ π (π+1 ,π‘ )π (π , π‘+β π‘ )=π (π ,π‘ )π (π , π‘+β π‘ )βπ (π , π‘ )=ΒΏβπ (π ,π‘ )=ΒΏ
+πͺ (β π‘ 2 )
m
m - 1
m + 1
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Master equations: Dynamics of population fractions
0 copies
Dt 2Dt0 t t + Dt
βπ (π , π‘ )β π‘ π πππ
=β [π β (π ,π‘ )+π β (π ,π‘ ) ] π (π , π‘ )π πππ
+π β (πβ1 ,π‘ ) π (πβ1 , π‘ )π πππ
+π β (π+1 ,π‘ ) π (π+1 , π‘ )ππππ
β [ π β (π , π‘ )+π β (π ,π‘ ) ]β π‘ π (π ,π‘ )+π β (πβ1 ,π‘ )β π‘ π (πβ1 ,π‘ )+π β (π+1 ,π‘ )β π‘ π (π+1 ,π‘ )βπ (π ,π‘ )=ΒΏ
+πͺ (β π‘ )
+πͺ (β π‘ 2 )
m
m - 1
m + 1
6
Master equations: Dynamics of population fractions
0 copies
Dt 2Dt0 t t + Dt
+πͺ (β π‘ )
βπ (π ,π‘ )β π‘ =β [π β (π , π‘ )+π β (π , π‘ ) ]π (π ,π‘ )+π β (πβ1 ,π‘ )π (πβ1 ,π‘ )+π β (π+1 , π‘ ) π (π+1 , π‘ )
βπ (π , π‘ )β π‘ π πππ
=β [π β (π ,π‘ )+π β (π ,π‘ ) ] π (π , π‘ )π πππ
+π β (πβ1 ,π‘ ) π (πβ1 , π‘ )π πππ
+π β (π+1 ,π‘ ) π (π+1 , π‘ )ππππ
+πͺ (β π‘ )ππ (π , π‘ )
ππ‘
m
m - 1
m + 1
Ensemble distributions Exact particular trajectories
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Stochastic modeling
Visualize master equation Walk through stochastic simulation script
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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
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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
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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
π₯=[π₯ (1 ) ΒΏ copies of mRNAπ₯ (2 ) ΒΏ copies of protein ]System variables:
System processes:
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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:
π₯=[π₯ (1 ) ΒΏ copies of mRNAπ₯ (2 ) ΒΏ copies of protein ]
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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:
π₯=[π₯ (1 ) ΒΏ copies of mRNAπ₯ (2 ) ΒΏ copies of protein ]
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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:
π₯=[π₯ (1 ) ΒΏ copies of mRNAπ₯ (2 ) ΒΏ copies of protein ]
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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
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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:
π₯=[π₯ (1 ) ΒΏ copies of mRNAπ₯ (2 ) ΒΏ copies of protein ]
Adding time-rates for individual channels
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a1
a2
a3
a4
a0
Reaction channel firings in a population
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a0
a0
a0
a0
a0
β π‘ β π π΄ππ πΈππΈππ β π0β π‘
a0
a0
a0
a0
a0
β π‘
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Reaction channel firings in a population
β π π΄ππ πΈππΈππ β π0β π‘
a0
a0
a0
a0
a0
β π‘
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β π π΄ππ πΈππΈππ β π0β π‘
Reaction channel firings in a population
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a0
a0
a0
a0
a0
Reaction channel firings in a population
β π π΄ππ πΈππΈππ β π0β π‘β π‘ 1 πβ π‘ 2 π
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a0
a0
a0
a0
a0
Reaction channel firings in a population
β π π΄ππ πΈππΈππ β π0β π‘β π‘ 1 πβ π‘ 2 π
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a0
a0
a0
a0
a0
β π‘πΏπππΊβ π‘πΏπππΊπΈπ
Reaction channel firings in a population
a0
a0
a0
a0
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a0
a0
a0
a0
a0
β π‘ππππ ππ»ππ π
β π‘ π΄πΏππ ππππππ»ππ π
Reaction channel firings in a populationZoomed in time scale
Dice represent shorter time intervals than before
β π π΄ππ πΈππΈππ β π0β π‘
Reaction channel firings in a population
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a0
a0
a0
a0
a0
β π‘ππ»ππ π β π π΄ππ πΈππΈππ β π0β π‘β π‘πΏπππΊ
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Draw duration from exponential distribution
β π π΄ππ πΈππΈππ β π0β π‘
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Draw duration from exponential distribution
β π π΄ππ πΈππΈππ β π0β π‘
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Draw duration from exponential distribution
β π π΄ππ πΈππΈππ β π0β π‘
28
Draw duration from exponential distribution
β π π΄ππ πΈππΈππ β π0β π‘
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Draw duration from exponential distribution
30
Draw duration from exponential distribution
31
Draw duration from exponential distribution
32
Draw duration from exponential distribution
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Draw duration until next event from exponential distribution
π=π‘0 ln( 1π 1 )
π 1=ΒΏ 0 1
π‘ 0β β¨π β©
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Draw duration from exponential distribution
π=π‘0 ln( 1π 1 )
π 1=ΒΏ 0 1
π‘ 0β β¨π β©
π=π‘0 ln( 1π 1 )
π 1=ΒΏ 0 1
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π=1.4 π‘0
Draw duration from exponential distribution
ΒΏ1.4 /π0
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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
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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