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C
k +
k −
B
A
C
BA
( ) ( )2
2
, , p x t p x t D
t x
∂ ∂=
∂ ∂
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A B
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x x x + ∆ x x −∆
12
12
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A B
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0 50 100 150 2000
0.5
1
1.5
time (ms)
Height
(nm)
0.8
0.60.4
0.2
50 100 150time (s)
0
B
B
δ
A B C
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x
t
( ),t ξ ω
( )time t
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( )time t
( )t ξ
( ) ( )1
t ξ ( ) ( )2
t ξ ( ) ( )3
t ξ
( )time t
0 T
( )time t
0 T
( )time t
0 T
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Bandwidth
A) B)
ω
ω
τ
τ
( ) x B τ
( ) y B τ
( ) xS ω
( ) yS ω
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R
C
T
( ) I t
( )Q t
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t
1 x
2 x
3 x
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1n − 1n +nsite site
1ng
− ng
1nr +n
r
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A)
ˆ x x
n=
1
2
( )ˆ x ρ
1
8n
B)
n
balls
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C
k +
k −
B
A
C
BA
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C ~ 10
C ~ 100
C ~ 1000
b)
time
a) c)
time
[ ]C [ ]C [ ]( )C t
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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
i p i p i p
in
in
in
10n = 50n = 100n =
B)A) C)
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b
a
Fixed Point isStable
Limit CycleAbout theFixed Point
1 1b a< +
1b a> +
2 X
1 X
1b a> + 2 X
1 X
1b a< +
a) b) c)
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1. Initialize: t← t0, n← n0.
2. Pick τ according to the density functiona,b
3.Pick µ according to the density functiona
4.Advance the process:•ni← ni + Siµ
• t← t + τ.c
5. Record as required for sampling or plotting.
If the process is to continue, then return to 2;
otherwise, stop.
( ) ( ) ( )1| , exp . p t a aτ τ = − n n n
( ) ( )
( )2 | , p t
a
µ
µ
ν ∆ =
nn n
n
( ), for ,
, for ;
t t t t
t t
µ τ ′− − < <
′ = ′ =
n Sn
n
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Number of X1 molecules (x103)
1 2 43 5
1
2
3
4
5
1 2 43 5
1
2
3
4
5
Number of X1 molecules (x103)
N u m b e r o f X 2
m o l e c u l e s ( x 1
0 3 )
N u m b e r o f X 2
m o l e c u l e s ( x 1
0 3 )
A B
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Number of X1 molecules (x103)1 2 43 5
1
2
3
4
5
N u m b e r o f X 2 m o l e c u l e s ( x
1 0 3 )
B
1
2
3
4
5
N u m b e r o f X j m o l e c u l e s ( x 1 0 3 )
A
1 2 3 4 5 6 7 8 9 10
Time
Number of X1
molecules (x103)
6 8 1210 14
4
6
8
12
14
Number of X1
molecules (x103)
N u m b e r o f X 2
m o l e c u l e s ( x 1 0 3 )
N u m b e r o f X 2 m o l e c u l e s ( x 1 0 3 )A B
10
2
4
6
8
12
14
10
2
2 4 6 8 1210 142 4
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Ω→∞
Large Volume( ) x t
( ), t α Π
a) b)
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1n
1n
2n
2n
a) b)
1000 2000 3000
1000
2000
3000
1000 2000 3000
1000
2000
3000
1n
2n
1000 2000 3000
1000
2000
3000
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Transcription
Translation
p
m
mα
pα
p β m
β
A) B)
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P r o t e i n n u m b
e r
0 100 200 300 400 5000
100
200
300
400
500
600
Time (min) Time (min)
P r o t e i n n u m b
e r
b 15
25
20
10
5
50 100 150
A) B)
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( ) ( ) ( ) ( )3 3 1 1 3 3 2 2 2 2 1 1 2 1 2 3, | , , | , , | , f x t x t f x t x t f x t x t dx t t t
∞
−∞
= < <∫
Chapman-Kolmogorov Equation - A (nonlinear) functional equationgoverning all conditional probability densities of a Markov process.
StationaryProcess
( ) ( ) ( )3 1 3 2 2 1 2| , | , | , p x x p x x p x x dxτ τ τ τ
∞
−∞
′ ′+ = ∫
Chapman-Kolmogorov Equation
( ) ( )2
212
p A y p B y pt y y
∂ ∂ ∂= − + ∂ ∂ ∂
Assumean=0, n>2
Kolmogorov Equation
Compute frommicroscopic theory:
( ) ( ) ( )
( ) ( )
0| , 1
|
p x z a x z
w x z o
τ τ δ
τ τ
′ ′= − −
′ ′+ +
( ) ( ) ( ) ( ) ( )3 1 3 2 2 1 2 3 3 1 2| , | | , | | , p x x w x x p x x w x x p x x dxτ τ τ
τ
∞
−∞
∂= −
∂ ∫
M ast er Equ at ion
Assume the jumps are “small”, and p( y,t ) is a slowly varying function of y.
( ) ( )
( ) ( )
2
1 22
1
2
,n
n
p
a y p a y pt y y
a y z w y z dz
∞
−∞
∂ ∂ ∂
= − + ∂ ∂ ∂
= ∫
Fokker-PlanckEquation
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0 y ( ) y t eq
y
0t >
( ), p y t ( )eq p y
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t t t
y y y
( ),P y t ( ),P y t ( ),P y t
( )0 y yδ −
( )
( )
where y t
dy A y
dt = ( ) 0
y t y A t = + ⋅
Width
2 D t ⋅
( ) 0
k t y t y e
− ⋅
=
( ) ( )2
Width
2 / 2 1 k t
D k e− ⋅
⋅ −
A) B) C)
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A) B)
xA
B
C
( )U x
0 x
W
A
Aω B
exp W
kT
−
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θ
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1. Initialize: t← t0, y← y0.
2. Choose a suitably small ∆t>0.a
3. Draw a sample value n of the unitnormal random variable N(0,1).b
4.Advance the process:
•y← y + n· c(y,t) [∆t]1/2 + A(y,t)∆t• t← t + ∆t.c
5. Record y(t)=y as required forsampling or plotting. If the process isto continue, then return to 2 or 3;d
otherwise, stop.
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t<<τcατc>>1Static Averaging
t>>τcατc<<1Kubo Renormalization
ατc<<1Bourret Convolution
αt<<1ατc>1
α2τct<<1ατc<1Born Iteration
Condition on t Condition onKubo number
Approximation
or
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( ) ( ) ( ) ( )0,1 x t t x t x t t + ∆ = − ⋅ ∆ ⋅N
( ) ( ) ( ) ( )
( ) ( ) ( )2 11 0,1
2
exp
c
c
x t t x t x t t t
t t t
t
η
η ρ η ρ τ
ρ τ
+ ∆ = − ⋅ ∆
+ ∆ = ⋅ + −
∆≡ −
N
1.
2.1 2 3 4 5
2
4
6
8
10
121.
2.
( ) x t
time
B.A.
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A) B)
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Noise – Induced
Oscillations
( )110 Ab
−∆ ×
R
A
δ
δ
0.25 0.5 0.75 1
0.02
0.04
0.06
0.08
0.10
0.12
0.14Oscillations
Monostable
X
1 2 3 4 5 6 7
500
1000
1500
( ) R t
days
A) B)
0λ ′ =
0λ =
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A
A
γ δ [ ]( )ln g A
[ ]ln A
AK
AK
f
A B C
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λ N
µ N
Transport between subvolumes:
( ) ( )1, , 1Tr
iw N
i i i i N N N N
λ µλ λ µ λ µ
+ → +
Each subvolume is a
spatially-homogeneous
reaction vessel
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0 0 1 0
1 1 1 1 0 0 1 0
0 0 0 1 1 1 1 1 0 0 1 0
0 0 0 1 1 1 1 1
0 0 0 1
− − −
− − −
− − −
1 2 3 4
λ λ λ λ ν ν ν ν
1 1 1 1
1 2 3 4
λ λ λ λ ν ν ν ν
+ + + +1 1 1 1
1 2 3 4
λ λ λ λ ν ν ν ν
− − − −
1
1
n
n
n
λ
λ
λ
−
+
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1 1 1 1 0 0 1 0 0 0 0 1
0 0 0 1 1 1 1 1 0 0 1 0
0 0 1 0 0 0 0 1 1 1 1 1
− − −
− − −
− − −
1
2
3
n
n
n
Transport from n1 to n3
Transport from n3 to n
1
S =
1 1 0 1 0 0 1 0 0 0 0 0
0 0 0 1 1 1 1 1 0 0 1 0
0 0 0 0 0 0 0 1 1 1 1 0
− −
− − −
− −
1
2
3
n
n
n
No transport to the left
S =
No transport to the right
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position, x
50
15.02
=
=
c
Dt
0 0.2 0.4 0.6 0.8 10
50
100
150
200
c o n c
e n t r a t i o n
B
0 0.2 0.4 0.6 0.8 10
50
100
150
200
position, x
c o n c
e n t r a t i o n
A
50
1.02
=
=
c
Dt
0 0.2 0.4 0.6 0.8 10
50
100
150
200
position, x
c o n c e
n t r a t i o n
C
50
2.02
=
=
c
Dt
0 0.2 0.4 0.6 0.8 10
50
100
150
200
position, x
c o n c e
n t r a t i o n
D
50
12
=
=
c
Dt
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0 0.2 0.4 0.6 0.8 10
50
100
150
200
0 0.2 0.4 0.6 0.8 10
100
200
300
400
0 0.2 0.4 0.6 0.8 10
10
20
30
40
position, x
c o n c e n t r a t i o n
position, x position, x
A B C
2 1
50
Dt
c
=
=
2 1
100
Dt
c
=
=
2 1
10
Dt
c
=
=
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0 0.2 0.4 0.6 0.8 10
50
100
150
200
250
c o n c e n t r a t i o n
position, x
A
0 0.2 0.4 0.6 0.8 10
1020
30
40
50
60
c o n c e n t r a t i o n
position, x
B
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
Uncertainty inthe interface
position is lessthan 4%
c o n c e n t r a t i o n
position, x
C Uncertainty inthe high stateis about 25%
Target molecule:
AverageStandarddeviation
Simulation
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0.5 1 1.5 2
2
4
6
8
10
Stable
Turing Unstable
NoiseNoise--inducedinducedTuringTuring--likelike
instabilityinstability
H D
H ρ Inhibitor degradation rate,
Ratio of
diffusivity Unstable
a
b
d
c
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Temporal instabili ty Turing instabili tyNoise-induced
spatial patterningSpatial patterning in
activator aloneA B C D
30
20
10
40 80 120 160 200
p o s i t i o n
time
00
200 600 1000 1400
30
20
10
40 80 120 160 200
p o s i t i o n
time
00
100 200 300 500
30
20
10
40 80 120 160 200
p o s i t i o n
time
00
100 200 300 400
30
20
10
40 80 120 160 200
p o s i t i o n
time
00
100 200 300400
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0 0.2 0.4 0.6 0.8 10
50
100
Wavenumber, k
( ) HH S k
inhibitor
0 0.2 0.4 0.6 0.8 10
50
100
Wavenumber, k
( ) HH S k
inhibitor
0 0.2 0.4 0.6 0.8 10
50
100
Wavenumber, k
( ) AAS k activator
A
0 0.2 0.4 0.6 10
20
40
0.8
( ) AAS k activator
B
Wavenumber, k
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E
EE
E
E
E P P
P
1 volt
1
1
1
1
.824 .785
.876 .503 .317 0
00
1
1
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Macroscopiclimit cycle
1n
2
n s
r
φ
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a)
2000 4000 6000 8000 10000
2000
4000
6000
8000
10000
1n
2n
2000 4000 6000 8000 10000
2000
4000
6000
8000
10000b)
1n
2n
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c− c+
0U
W −
W +
( )U x
x
Variance of white noise forcing
Signal-to-noise ratio
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Kalman
Filter
LeastSquares
StochasticProcesses
ProbabilityTheory
DynamicalSystems
LinearAlgebra
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( )nF x
( )nF x
−
( )F x
xn
x
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( ) xθ
x0
1
( )
d x
dx
θ δ =
x0
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A) B)
C) D)
( )baU X ,=
x
( ) f x
ab −
1
x
( ) X E a=
( ) f x
a1
a e
a
µ−σ µ µ+σ
2σ( )2, X N µ σ =
x
( ) f x
22
1
πσ
µ−σ µ µ+σ
2σ
( ), X C µ σ =
x
( ) f x
1πσ
( ) ( )
1
0 otherwise
b a a x b f x
− − ≤ ≤=
( ) [ ]exp 0 f x a ax x= − ≥
( ) ( )
2
22
1exp
22
x f x
µ
σ πσ
−= −
( ) ( )
( )2 2
f x x
σ π
µ σ
=− +
a b
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A) B)
C) D)
0.2
0.4
0.6
0.8
1
x
( ) xF
( )baU X ,=
µ-σ µ
0.2
0.4
0.6
0.8
1
µ+σ x
( ) xF
( )2, X N µ σ =
0.2
0.4
0.6
0.8
µ-σ µ µ+σ x
( ) xF
( ), X C µ σ =
0.2
0.4
0.6
0.8
1
x
( ) xF
( ) X E a=
( ) a x
F xa b
−=
−
( ) 1 ax
F x e−
= −
( )2
11
2 2
xF x erf
µ
σ
−= −
( ) 1 1
arctan2
xF x
µ
π σ
− = −
a b
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0.0625001/24
0.062450107
0.062643106
0.062140105
0.063300104
0.061000103
Estimate ofprobability to
flip 4 heads
Number oftrials, N
N=input('How many trials?');
number_of_4_heads=0;for i=1:N
f1=rand;f2=rand;
f3=rand;f4=rand;if f1>0.5 && f2>0.5 && f3>0.5 && f4>0.5
number_of_4_heads=number_of_4_heads+1;
endend
sprintf('Average number of 4 heads tossed:%0.6f',number_of_4_heads/N)
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0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1 y
x
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0.785398π /4
0.785627107
0.785315106
0.783310105
0.786300104
0.805000103
Estimate ofarea of quarter
circle
Number of
points, NN=input('How many points?');
pts_inside=0;for i=1:N
p1=rand;p2=rand;if p1^2+p2^2 <= 1
pts_inside=pts_inside+1;end
end
sprintf('Area of the quarter circle: %0.6f',pts_inside/N)
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P
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a
λ
d
r
θ x
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