calculus of biochemical noise
TRANSCRIPT
![Page 1: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/1.jpg)
Calculus of biochemical noise
Michał Komorowski
Institute of Fundamental Technological ResearchPolish Academy of Sciences
02/07/11
Michał Komorowski Calculus of biochemical noise 02/07/11 1 / 18
![Page 2: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/2.jpg)
Outline
1 Statistical analysis: Inference & sensitivity
2 Noise decomposition
3 Role of protein degradation
Michał Komorowski Calculus of biochemical noise 02/07/11 2 / 18
![Page 3: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/3.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...r
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fr(x,Θ))
ParametersΘ = (θ1, ..., θl)
x is a Poisson birth and death process
Michał Komorowski Calculus of biochemical noise Model 02/07/11 3 / 18
![Page 4: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/4.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...r
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fr(x,Θ))
ParametersΘ = (θ1, ..., θl)
x is a Poisson birth and death process
Michał Komorowski Calculus of biochemical noise Model 02/07/11 3 / 18
![Page 5: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/5.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...r
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fr(x,Θ))
ParametersΘ = (θ1, ..., θl)
x is a Poisson birth and death process
Michał Komorowski Calculus of biochemical noise Model 02/07/11 3 / 18
![Page 6: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/6.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...r
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fr(x,Θ))
ParametersΘ = (θ1, ..., θl)
x is a Poisson birth and death process
Michał Komorowski Calculus of biochemical noise Model 02/07/11 3 / 18
![Page 7: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/7.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...r
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fr(x,Θ))
ParametersΘ = (θ1, ..., θl)
x is a Poisson birth and death process
Michał Komorowski Calculus of biochemical noise Model 02/07/11 3 / 18
![Page 8: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/8.jpg)
Model equations
x(t) =
x(0) +
r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)
Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 9: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/9.jpg)
Model equations
x(t) = x(0)
+
r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)
Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 10: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/10.jpg)
Model equations
x(t) = x(0) +
r∑
j=1
S·j
Yj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)
Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 11: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/11.jpg)
Model equations
x(t) = x(0) +
r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)
Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 12: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/12.jpg)
Model equations
x(t) = x(0) +
r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)
Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 13: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/13.jpg)
Model equations
x(t) = x(0) +
r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)
Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 14: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/14.jpg)
Model equations
x(t) = x(0) +
r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +
r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)
Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 15: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/15.jpg)
Model equations
x(t) = x(0) +
r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)
Ball et al., Ann Appl Probab (2006).
S·j Change in the state due to occurrence of jth reaction
Y·(u) is a Poisson point process (How many times did reaction j fire?)
Y·(u) can be approximated as
Y(u) ≈ u + N(0, u)
Deterministic approximation
φ(t) = φ(0) +
r∑
j=1
S·j(∫ t
0fj(φ, s)ds)
dφdt
= S F(φ)Michał Komorowski Calculus of biochemical noise Model 02/07/11 4 / 18
![Page 16: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/16.jpg)
Model equations
ξ(t) ≡ x(t)− φ(t) =
= x(0)− φ(0) +r∑
j=1
S·j
∫ t
0fj(x(s), s)ds−
∫ t
0fj(φ(s), s)ds
∇φ∫ t
0fj(φ(s), s)ξ(s)ds
+r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)−∫ t
0fj(x(s), s)ds
Wj(
∫ t
0fj(φ(s), s)ds)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
Michał Komorowski Calculus of biochemical noise Model 02/07/11 5 / 18
![Page 17: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/17.jpg)
Model equations
ξ(t) ≡ x(t)− φ(t) =
= x(0)− φ(0) +
r∑
j=1
S·j
∫ t
0fj(x(s), s)ds−
∫ t
0fj(φ(s), s)ds
∇φ∫ t
0fj(φ(s), s)ξ(s)ds
+r∑
j=1
S·jYj
(∫ t
0fj(x, s)ds
)−∫ t
0fj(x(s), s)ds
Wj(
∫ t
0fj(φ(s), s)ds)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
Michał Komorowski Calculus of biochemical noise Model 02/07/11 5 / 18
![Page 18: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/18.jpg)
Model equations
ξ(t) ≡ x(t)− φ(t) =
= x(0)− φ(0) +
r∑
j=1
S·j
∫ t
0fj(x(s), s)ds−
∫ t
0fj(φ(s), s)ds
︸ ︷︷ ︸
∇φ∫ t
0fj(φ(s), s)ξ(s)ds
+r∑
j=1
S·j Yj
(∫ t
0fj(x, s)ds
)−∫ t
0fj(x(s), s)ds
︸ ︷︷ ︸
Wj(
∫ t
0fj(φ(s), s)ds)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
Michał Komorowski Calculus of biochemical noise Model 02/07/11 5 / 18
![Page 19: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/19.jpg)
Model equations
ξ(t) ≡ x(t)− φ(t) =
= x(0)− φ(0) +
r∑
j=1
S·j
∫ t
0fj(x(s), s)ds−
∫ t
0fj(φ(s), s)ds
︸ ︷︷ ︸
∇φ∫ t
0fj(φ(s), s)ξ(s)ds
+r∑
j=1
S·j Yj
(∫ t
0fj(x, s)ds
)−∫ t
0fj(x(s), s)ds
︸ ︷︷ ︸
Wj(
∫ t
0fj(φ(s), s)ds)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
Michał Komorowski Calculus of biochemical noise Model 02/07/11 5 / 18
![Page 20: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/20.jpg)
Model equations
ξ(t) ≡ x(t)− φ(t) =
= x(0)− φ(0) +
r∑
j=1
S·j
∫ t
0fj(x(s), s)ds−
∫ t
0fj(φ(s), s)ds
︸ ︷︷ ︸
∇φ∫ t
0fj(φ(s), s)ξ(s)ds
+r∑
j=1
S·j Yj
(∫ t
0fj(x, s)ds
)−∫ t
0fj(x(s), s)ds
︸ ︷︷ ︸
Wj(
∫ t
0fj(φ(s), s)ds)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
Michał Komorowski Calculus of biochemical noise Model 02/07/11 5 / 18
![Page 21: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/21.jpg)
Model equations
x(t) = φ(t) + ξ(t)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
LNA implies Gaussian distribution
x(t) ∼ MVN(φ(t),Σ(t))
Mean φ(t) given as s solution of the rate equationVariances
dΣ(t)dt
= A(φ, t)Σ + ΣA(φ, t)T + E(φ, t)E(φ, t)T
Covariances
cov(x(s), x(t)) = Σ(s)Φ(s, t)T for s ≥ t
dΦ(t, s)ds
= A(φ, s)Φ(t, s), Φ(t, t) = I
Michał Komorowski Calculus of biochemical noise Model 02/07/11 6 / 18
![Page 22: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/22.jpg)
Model equations
x(t) = φ(t) + ξ(t)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
LNA implies Gaussian distribution
x(t) ∼ MVN(φ(t),Σ(t))
Mean φ(t) given as s solution of the rate equationVariances
dΣ(t)dt
= A(φ, t)Σ + ΣA(φ, t)T + E(φ, t)E(φ, t)T
Covariances
cov(x(s), x(t)) = Σ(s)Φ(s, t)T for s ≥ t
dΦ(t, s)ds
= A(φ, s)Φ(t, s), Φ(t, t) = I
Michał Komorowski Calculus of biochemical noise Model 02/07/11 6 / 18
![Page 23: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/23.jpg)
Model equations
x(t) = φ(t) + ξ(t)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
LNA implies Gaussian distribution
x(t) ∼ MVN(φ(t),Σ(t))
Mean φ(t) given as s solution of the rate equationVariances
dΣ(t)dt
= A(φ, t)Σ + ΣA(φ, t)T + E(φ, t)E(φ, t)T
Covariances
cov(x(s), x(t)) = Σ(s)Φ(s, t)T for s ≥ t
dΦ(t, s)ds
= A(φ, s)Φ(t, s), Φ(t, t) = I
Michał Komorowski Calculus of biochemical noise Model 02/07/11 6 / 18
![Page 24: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/24.jpg)
Model equations
x(t) = φ(t) + ξ(t)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
LNA implies Gaussian distribution
x(t) ∼ MVN(φ(t),Σ(t))
Mean φ(t) given as s solution of the rate equationVariances
dΣ(t)dt
= A(φ, t)Σ + ΣA(φ, t)T + E(φ, t)E(φ, t)T
Covariances
cov(x(s), x(t)) = Σ(s)Φ(s, t)T for s ≥ t
dΦ(t, s)ds
= A(φ, s)Φ(t, s), Φ(t, t) = I
Michał Komorowski Calculus of biochemical noise Model 02/07/11 6 / 18
![Page 25: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/25.jpg)
Model equations
x(t) = φ(t) + ξ(t)
dξ = SOφF(φ)ξdt + S(
diag{√
F(φ)})
dW
LNA implies Gaussian distribution
x(t) ∼ MVN(φ(t),Σ(t))
Mean φ(t) given as s solution of the rate equationVariances
dΣ(t)dt
= A(φ, t)Σ + ΣA(φ, t)T + E(φ, t)E(φ, t)T
Covariances
cov(x(s), x(t)) = Σ(s)Φ(s, t)T for s ≥ t
dΦ(t, s)ds
= A(φ, s)Φ(t, s), Φ(t, t) = I
Michał Komorowski Calculus of biochemical noise Model 02/07/11 6 / 18
![Page 26: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/26.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 27: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/27.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 28: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/28.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 29: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/29.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 30: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/30.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 31: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/31.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 32: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/32.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 33: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/33.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 34: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/34.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 35: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/35.jpg)
Approximate Bayesian Computation
θ1
θ2
Model
t
X(t)
Data, X
Simulation, Xs(θ)
d = ∆(Xs(θ),X)
Reject θ if d > εAccept θ if d ≤ ε
Toni et al., J.Roy.Soc. Interface (2009).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 7 / 18
![Page 36: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/36.jpg)
ABC SMC
Prior, π(θ) Define set of intermediate distributions, πt, t = 1, ...., Tε1 > ε2 > ...... > εT
πt−1(θ|∆(Xs,X) < εt−1)
πt(θ|∆(Xs,X) < εt)
πT(θ|∆(Xs,X) < εT)
Sequential importance sampling:Sample from proposal, ηt(θt) and weightwt(θt) = πt(θt)/ηt(θt) withηt(θt) =
∫πt−1(θt−1)Kt(θt−1, θt)dθt−1 where
Kt(θt−1, θt) is Markov perturbation kernel
Toni et al., J.Roy.Soc. Interface (2009); Toni & Stumpf, Bioinformatics (2010).
Michał Komorowski Calculus of biochemical noise ABC 02/07/11 8 / 18
![Page 37: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/37.jpg)
Inference and Model selectionWe have observed data, D, that was generated by some system of in generalunknown structure that we seek to describe by a mathematical model. Inprinciple we can have a model-set,M = {M1, . . . ,Mν}, where each model Mi
has an associated parameter θi.
Model Posterior︷ ︸︸ ︷Pr(Mi|D) =
Likelihood︷ ︸︸ ︷Pr(D|Mi)
Prior︷ ︸︸ ︷π(Mi)
ν∑
j=1
Pr(D|Mj)π(Mj)
︸ ︷︷ ︸Evidence
For complicated modelsand/or detailed data thelikelihood evaluation canbecome prohibitivelyexpensive.
Approximate InferenceWe can approximate the models. The “true” model is unlikely to be inM anyway.Komorowski et al., BMC Bioinformatics (2009); Komorowski et al., Biophysical J. (2010)
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 9 / 18
![Page 38: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/38.jpg)
Inference and Model selectionWe have observed data, D, that was generated by some system of in generalunknown structure that we seek to describe by a mathematical model. Inprinciple we can have a model-set,M = {M1, . . . ,Mν}, where each model Mi
has an associated parameter θi.
Model Posterior︷ ︸︸ ︷Pr(Mi|D) =
Likelihood︷ ︸︸ ︷Pr(D|Mi)
Prior︷ ︸︸ ︷π(Mi)
ν∑
j=1
Pr(D|Mj)π(Mj)
︸ ︷︷ ︸Evidence
For complicated modelsand/or detailed data thelikelihood evaluation canbecome prohibitivelyexpensive.
Approximate InferenceWe can approximate the models. The “true” model is unlikely to be inM anyway.Komorowski et al., BMC Bioinformatics (2009); Komorowski et al., Biophysical J. (2010)
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 9 / 18
![Page 39: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/39.jpg)
Inference and Model selectionWe have observed data, D, that was generated by some system of in generalunknown structure that we seek to describe by a mathematical model. Inprinciple we can have a model-set,M = {M1, . . . ,Mν}, where each model Mi
has an associated parameter θi.
Model Posterior︷ ︸︸ ︷Pr(Mi|D) =
Likelihood︷ ︸︸ ︷Pr(D|Mi)
Prior︷ ︸︸ ︷π(Mi)
ν∑
j=1
Pr(D|Mj)π(Mj)
︸ ︷︷ ︸Evidence
For complicated modelsand/or detailed data thelikelihood evaluation canbecome prohibitivelyexpensive.
Approximate InferenceWe can approximate the models. The “true” model is unlikely to be inM anyway.Komorowski et al., BMC Bioinformatics (2009); Komorowski et al., Biophysical J. (2010)
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 9 / 18
![Page 40: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/40.jpg)
Inference and Model selectionWe have observed data, D, that was generated by some system of in generalunknown structure that we seek to describe by a mathematical model. Inprinciple we can have a model-set,M = {M1, . . . ,Mν}, where each model Mi
has an associated parameter θi.
Model Posterior︷ ︸︸ ︷Pr(Mi|D) =
Likelihood︷ ︸︸ ︷Pr(D|Mi)
Prior︷ ︸︸ ︷π(Mi)
ν∑
j=1
Pr(D|Mj)π(Mj)
︸ ︷︷ ︸Evidence
For complicated modelsand/or detailed data thelikelihood evaluation canbecome prohibitivelyexpensive.
Approximate InferenceWe can approximate the models. The “true” model is unlikely to be inM anyway.Komorowski et al., BMC Bioinformatics (2009); Komorowski et al., Biophysical J. (2010)
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 9 / 18
![Page 41: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/41.jpg)
Inference and Model selectionWe have observed data, D, that was generated by some system of in generalunknown structure that we seek to describe by a mathematical model. Inprinciple we can have a model-set,M = {M1, . . . ,Mν}, where each model Mi
has an associated parameter θi.
Model Posterior︷ ︸︸ ︷Pr(Mi|D) =
Likelihood︷ ︸︸ ︷Pr(D|Mi)
Prior︷ ︸︸ ︷π(Mi)
ν∑
j=1
Pr(D|Mj)π(Mj)
︸ ︷︷ ︸Evidence
For complicated modelsand/or detailed data thelikelihood evaluation canbecome prohibitivelyexpensive.
Approximate InferenceWe can approximate the models. The “true” model is unlikely to be inM anyway.Komorowski et al., BMC Bioinformatics (2009); Komorowski et al., Biophysical J. (2010)
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 9 / 18
![Page 42: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/42.jpg)
Parameter IdentifiabilityParameter Identifiability
f (Θ)f (Θ �)
t
y(t)
θ1
θ2 Sloppy
Stiff
Fisher Information
I(Θ)k ,l = EΘ
��∂
∂θklog(Pr(D|Θ))
��∂
∂θllog(Pr(D|Θ))
��
≈ ∂µ
∂θk
T
Σ(Θ)∂µ
∂θl+
12
trace(Σ−1 ∂Σ
∂θkΣ−1 ∂Σ
∂θl)in the LNA.
Erguler & Stumpf, MolBiosyst. (2011); Komorowski et al., PNAS (2011).
Inference Based Modelling Michael P.H. Stumpf Parameter Estimation 5 of 26
θ1
θ2 Sloppy
Stiff
Fisher Information
I(Θ)k,l = EΘ
[(∂
∂θklog(Pr(D|Θ))
)(∂
∂θllog(Pr(D|Θ))
)]
≈ ∂µ
∂θk
TΣ(Θ)
∂µ
∂θl+
12
trace(Σ−1 ∂Σ
∂θkΣ−1∂Σ
∂θl) in the LNA.
Komorowski et al., PNAS (2011).
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 10 / 18
![Page 43: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/43.jpg)
Parameter IdentifiabilityParameter Identifiability
f (Θ)f (Θ �)
t
y(t)
θ1
θ2 Sloppy
Stiff
Fisher Information
I(Θ)k ,l = EΘ
��∂
∂θklog(Pr(D|Θ))
��∂
∂θllog(Pr(D|Θ))
��
≈ ∂µ
∂θk
T
Σ(Θ)∂µ
∂θl+
12
trace(Σ−1 ∂Σ
∂θkΣ−1 ∂Σ
∂θl)in the LNA.
Erguler & Stumpf, MolBiosyst. (2011); Komorowski et al., PNAS (2011).
Inference Based Modelling Michael P.H. Stumpf Parameter Estimation 5 of 26
θ1
θ2 Sloppy
Stiff
Fisher Information
I(Θ)k,l = EΘ
[(∂
∂θklog(Pr(D|Θ))
)(∂
∂θllog(Pr(D|Θ))
)]
≈ ∂µ
∂θk
TΣ(Θ)
∂µ
∂θl+
12
trace(Σ−1 ∂Σ
∂θkΣ−1∂Σ
∂θl) in the LNA.
Komorowski et al., PNAS (2011).
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 10 / 18
![Page 44: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/44.jpg)
Parameter IdentifiabilityParameter Identifiability
f (Θ)f (Θ �)
t
y(t)
θ1
θ2 Sloppy
Stiff
Fisher Information
I(Θ)k ,l = EΘ
��∂
∂θklog(Pr(D|Θ))
��∂
∂θllog(Pr(D|Θ))
��
≈ ∂µ
∂θk
T
Σ(Θ)∂µ
∂θl+
12
trace(Σ−1 ∂Σ
∂θkΣ−1 ∂Σ
∂θl)in the LNA.
Erguler & Stumpf, MolBiosyst. (2011); Komorowski et al., PNAS (2011).
Inference Based Modelling Michael P.H. Stumpf Parameter Estimation 5 of 26
θ1
θ2 Sloppy
Stiff
Fisher Information
I(Θ)k,l = EΘ
[(∂
∂θklog(Pr(D|Θ))
)(∂
∂θllog(Pr(D|Θ))
)]
≈ ∂µ
∂θk
TΣ(Θ)
∂µ
∂θl+
12
trace(Σ−1 ∂Σ
∂θkΣ−1∂Σ
∂θl) in the LNA.
Komorowski et al., PNAS (2011).
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 10 / 18
![Page 45: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/45.jpg)
Inferability and Fisher Information
Komorowski et al., PNAS (2011).
Michał Komorowski Calculus of biochemical noise Inference 02/07/11 11 / 18
![Page 46: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/46.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 47: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/47.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 48: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/48.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 49: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/49.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 50: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/50.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 51: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/51.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 52: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/52.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 53: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/53.jpg)
Who are noisemakers?How much noise is generated by each of the reactions in a givensystem?How does noise enter biochemical system and is propagatedthrough reaction systems?
x(t) = x(0) +
r∑
j=1
S·jYj(
∫ t
0fj(x, s)ds)
Defining noise contribution of each reactionProcess with averaged timings of a single reaction
〈x(t)〉|−j
x(t)− 〈x(t)〉|−j
Variance contribution of individual reactions
Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 12 / 18
![Page 54: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/54.jpg)
Noise decomposition
Using linear noise approximation:
Variance decomposition
Σ(t) = Σ(1)(t) + ... + Σ(r)(t)
dΣ
dt= A(t)Σ + ΣA(t)T + D(t)
dΣ(j)
dt= A(t)Σ(j) + Σ(j)A(t)T + D(j)(t)
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 13 / 18
![Page 55: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/55.jpg)
Noise decomposition
Using linear noise approximation:
Variance decomposition
Σ(t) = Σ(1)(t) + ... + Σ(r)(t)
dΣ
dt= A(t)Σ + ΣA(t)T + D(t)
dΣ(j)
dt= A(t)Σ(j) + Σ(j)A(t)T + D(j)(t)
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 13 / 18
![Page 56: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/56.jpg)
Noise decomposition
Using linear noise approximation:
Variance decomposition
Σ(t) = Σ(1)(t) + ... + Σ(r)(t)
dΣ
dt= A(t)Σ + ΣA(t)T + D(t)
dΣ(j)
dt= A(t)Σ(j) + Σ(j)A(t)T + D(j)(t)
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 13 / 18
![Page 57: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/57.jpg)
Birth and death process
∅ k+−→ x
x k−−→ ∅How much of the noise comes from birth and how much comesfrom death?
dx = (k+ − k−x(t))dt +√
k+dW1︸ ︷︷ ︸birth noise
+√
k−〈x(t)〉dW2︸ ︷︷ ︸death noise
Σ =12〈x〉︸︷︷︸
birth noise
+12〈x〉︸︷︷︸
death noise
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 14 / 18
![Page 58: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/58.jpg)
Birth and death process
∅ k+−→ x
x k−−→ ∅How much of the noise comes from birth and how much comesfrom death?
dx = (k+ − k−x(t))dt +√
k+dW1︸ ︷︷ ︸birth noise
+√
k−〈x(t)〉dW2︸ ︷︷ ︸death noise
Σ =12〈x〉︸︷︷︸
birth noise
+12〈x〉︸︷︷︸
death noise
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 14 / 18
![Page 59: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/59.jpg)
Birth and death process
∅ k+−→ x
x k−−→ ∅How much of the noise comes from birth and how much comesfrom death?
dx = (k+ − k−x(t))dt +√
k+dW1︸ ︷︷ ︸birth noise
+√
k−〈x(t)〉dW2︸ ︷︷ ︸death noise
Σ =12〈x〉︸︷︷︸
birth noise
+12〈x〉︸︷︷︸
death noise
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 14 / 18
![Page 60: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/60.jpg)
Birth and death process
∅ k+−→ x
x k−−→ ∅How much of the noise comes from birth and how much comesfrom death?
dx = (k+ − k−x(t))dt +√
k+dW1︸ ︷︷ ︸birth noise
+√
k−〈x(t)〉dW2︸ ︷︷ ︸death noise
Σ =12〈x〉︸︷︷︸
birth noise
+12〈x〉︸︷︷︸
death noise
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 14 / 18
![Page 61: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/61.jpg)
Noise contributions - general resultsTheorem 1In any open conversion system with only first order reactions thecontribution of the product degradation to the variability in the productabundance is precisely one half of the total variability.
[Σ(r)
]nn
=12
[Σ]nn
Theorem 2In a general system the contribution of the product degradation to thevariability in the product abundance is one half its mean.
[Σ(r)]nn =12〈xn〉
Komorowski M., Miekisz J., Stumpf M.P.H. Decomposing Noise in Biochemical Signalling Systems Highlights the Role of ProteinDegradation, Submitted, available on ArXiv.
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 15 / 18
![Page 62: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/62.jpg)
Noise contributions - general resultsTheorem 1In any open conversion system with only first order reactions thecontribution of the product degradation to the variability in the productabundance is precisely one half of the total variability.
[Σ(r)
]nn
=12
[Σ]nn
Theorem 2In a general system the contribution of the product degradation to thevariability in the product abundance is one half its mean.
[Σ(r)]nn =12〈xn〉
Komorowski M., Miekisz J., Stumpf M.P.H. Decomposing Noise in Biochemical Signalling Systems Highlights the Role of ProteinDegradation, Submitted, available on ArXiv.
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 15 / 18
![Page 63: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/63.jpg)
Noise contributions - general resultsTheorem 1In any open conversion system with only first order reactions thecontribution of the product degradation to the variability in the productabundance is precisely one half of the total variability.
[Σ(r)
]nn
=12
[Σ]nn
Theorem 2In a general system the contribution of the product degradation to thevariability in the product abundance is one half its mean.
[Σ(r)]nn =12〈xn〉
Komorowski M., Miekisz J., Stumpf M.P.H. Decomposing Noise in Biochemical Signalling Systems Highlights the Role of ProteinDegradation, Submitted, available on ArXiv.
Michał Komorowski Calculus of biochemical noise Noise decomposition 02/07/11 15 / 18
![Page 64: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/64.jpg)
Linear cascades
X1 X2 X31 2 3
4 5 6
X1 X2 X31 2 3
4 5 6
(A)
(B)
7%
6%
25%
3%
8%
50%
conversion slow
123456
36%
2%5%
3% 4%
50%
conversion fast
2%
17%
33%
2%
16%
31%
catalytic slow
20%
10%
20% 20%
10%
20%
catalytic fast
PropositionIn the catalytic linear pathways sum of contributions of productionreactions is equal to the sum of contributions of degradation reactions.
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 16 / 18
![Page 65: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/65.jpg)
Linear cascades
X1 X2 X31 2 3
4 5 6
X1 X2 X31 2 3
4 5 6
(A)
(B)
7%
6%
25%
3%
8%
50%
conversion slow
123456
36%
2%5%
3% 4%
50%
conversion fast
2%
17%
33%
2%
16%
31%
catalytic slow
20%
10%
20% 20%
10%
20%
catalytic fast
PropositionIn the catalytic linear pathways sum of contributions of productionreactions is equal to the sum of contributions of degradation reactions.
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 16 / 18
![Page 66: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/66.jpg)
Linear cascades
X1 X2 X31 2 3
4 5 6
X1 X2 X31 2 3
4 5 6
(A)
(B)
7%
6%
25%
3%
8%
50%
conversion slow
123456
36%
2%5%
3% 4%
50%
conversion fast
2%
17%
33%
2%
16%
31%
catalytic slow
20%
10%
20% 20%
10%
20%
catalytic fast
PropositionIn the catalytic linear pathways sum of contributions of productionreactions is equal to the sum of contributions of degradation reactions.
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 16 / 18
![Page 67: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/67.jpg)
Michaelis - Menten Kinetics
∅ kb−→ S
S + Ek0S·E−⇀↽−
k1C
Ck2−→ E + P
Pkd−→ ∅
42%
9% 8%
41%
substrate10%
40% 39%
11%
enzyme
10%
40% 39%
11%
complex
32%
< 1%< 1%
17%
50%
product
subtrate +complex forwardcomplex backwardproduct +prod. degradation
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 17 / 18
![Page 68: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/68.jpg)
Michaelis - Menten Kinetics
∅ kb−→ S
S + Ek0S·E−⇀↽−
k1C
Ck2−→ E + P
Pkd−→ ∅
42%
9% 8%
41%
substrate10%
40% 39%
11%
enzyme
10%
40% 39%
11%
complex
32%
< 1%< 1%
17%
50%
product
subtrate +complex forwardcomplex backwardproduct +prod. degradation
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 17 / 18
![Page 69: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/69.jpg)
Michaelis - Menten Kinetics
∅ kb−→ S
S + Ek0S·E−⇀↽−
k1C
Ck2−→ E + P
Pkd−→ ∅
42%
9% 8%
41%
substrate10%
40% 39%
11%
enzyme
10%
40% 39%
11%
complex
32%
< 1%< 1%
17%
50%
product
subtrate +complex forwardcomplex backwardproduct +prod. degradation
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 17 / 18
![Page 70: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/70.jpg)
Michaelis - Menten Kinetics
∅ kb−→ S
S + Ek0S·E−⇀↽−
k1C
Ck2−→ E + P
Pkd−→ ∅
42%
9% 8%
41%
substrate10%
40% 39%
11%
enzyme
10%
40% 39%
11%
complex
32%
< 1%< 1%
17%
50%
product
subtrate +complex forwardcomplex backwardproduct +prod. degradation
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 17 / 18
![Page 71: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/71.jpg)
Michaelis - Menten Kinetics
∅ kb−→ S
S + Ek0S·E−⇀↽−
k1C
Ck2−→ E + P
Pkd−→ ∅
42%
9% 8%
41%
substrate10%
40% 39%
11%
enzyme
10%
40% 39%
11%
complex
32%
< 1%< 1%
17%
50%
product
subtrate +complex forwardcomplex backwardproduct +prod. degradation
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 17 / 18
![Page 72: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/72.jpg)
Fluorescent protein maturation noise
12%
12%
4%
47%
25%
slow maturation
30%
30%
13%
23%
4%fast maturation
transcriptionRNA deg.translationtot. protein deg.fold. & mat.
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 18 / 18
![Page 73: Calculus of biochemical noise](https://reader033.vdocuments.site/reader033/viewer/2022052412/5588f224d8b42a356b8b4605/html5/thumbnails/73.jpg)
Acknowledgement
Michael StumpfImperial College London
Jacek MiekiszUniversity of Warsaw
Barbel FinkenstadWarwick University
David RandWarwick University
Michał Komorowski Calculus of biochemical noise Examples 02/07/11 18 / 18