Calculus of biochemical noise

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

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

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


System’s state

x = (x1, . . . , xN)T


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))





System’s state

x = (x1, . . . , xN)T


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))





System’s state

x = (x1, . . . , xN)T


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))





System’s state

x = (x1, . . . , xN)T


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))




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)


Model equations

x(t) = x(0)

+

r∑

j=1

S·jYj

(∫ t

0fj(x, s)ds

)





Y(u) ≈ u + N(0, u)


φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)

dφdt

= S F(φ)


Model equations

x(t) = x(0) +

r∑

j=1

S·j

Yj

(∫ t

0fj(x, s)ds

)





Y(u) ≈ u + N(0, u)


φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)

dφdt

= S F(φ)


Model equations

x(t) = x(0) +

r∑

j=1

S·jYj

(∫ t

0fj(x, s)ds

)





Y(u) ≈ u + N(0, u)


φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)

dφdt

= S F(φ)


Model equations

x(t) = x(0) +

r∑

j=1

S·jYj

(∫ t

0fj(x, s)ds

)





Y(u) ≈ u + N(0, u)


φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)

dφdt

= S F(φ)


Model equations

x(t) = x(0) +

r∑

j=1

S·jYj

(∫ t

0fj(x, s)ds

)





Y(u) ≈ u + N(0, u)


φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)

dφdt

= S F(φ)


Model equations

x(t) = x(0) +

r∑

j=1

S·jYj

(∫ t

0fj(x, s)ds

)





Y(u) ≈ u + N(0, u)


φ(t) = φ(0) +

r∑

j=1

S·j(∫ t

0fj(φ, s)ds)

dφdt

= S F(φ)


Model equations

x(t) = x(0) +

r∑

j=1

S·jYj

(∫ t

0fj(x, s)ds

)





Y(u) ≈ u + N(0, u)


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

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


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


+r∑

j=1

S·jYj

(∫ t

0fj(x, s)ds

)−∫ t

0fj(x(s), s)ds

Wj(

∫ t

0fj(φ(s), s)ds)


diag{√

F(φ)})

dW


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


+r∑

j=1

S·j Yj

(∫ t

0fj(x, s)ds

)−∫ t

0fj(x(s), s)ds

︸︷︷︸

Wj(

∫ t

0fj(φ(s), s)ds)


diag{√

F(φ)})

dW


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


+r∑

j=1

S·j Yj

(∫ t

0fj(x, s)ds

)−∫ t

0fj(x(s), s)ds

︸︷︷︸

Wj(

∫ t

0fj(φ(s), s)ds)


diag{√

F(φ)})

dW


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


+r∑

j=1

S·j Yj

(∫ t

0fj(x, s)ds

)−∫ t

0fj(x(s), s)ds

︸︷︷︸

Wj(

∫ t

0fj(φ(s), s)ds)


diag{√

F(φ)})

dW


Model equations

x(t) = φ(t) + ξ(t)


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


Model equations

x(t) = φ(t) + ξ(t)


diag{√

F(φ)})

dW




dΣ(t)dt


Covariances


dΦ(t, s)ds

= A(φ, s)Φ(t, s), Φ(t, t) = I


Model equations

x(t) = φ(t) + ξ(t)


diag{√

F(φ)})

dW




dΣ(t)dt


Covariances


dΦ(t, s)ds

= A(φ, s)Φ(t, s), Φ(t, t) = I


Model equations

x(t) = φ(t) + ξ(t)


diag{√

F(φ)})

dW




dΣ(t)dt


Covariances


dΦ(t, s)ds

= A(φ, s)Φ(t, s), Φ(t, t) = I


Model equations

x(t) = φ(t) + ξ(t)


diag{√

F(φ)})

dW




dΣ(t)dt


Covariances


dΦ(t, s)ds

= A(φ, s)Φ(t, s), Φ(t, t) = I


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


θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)





θ1

θ2

Model

t

X(t)

Data, X

Simulation, Xs(θ)

d = ∆(Xs(θ),X)




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).


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






ν∑

j=1

Pr(D|Mj)π(Mj)










ν∑

j=1

Pr(D|Mj)π(Mj)










ν∑

j=1

Pr(D|Mj)π(Mj)










ν∑

j=1

Pr(D|Mj)π(Mj)





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).



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.



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.




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.



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.



Inferability and Fisher Information



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


x(t) = x(0) +

r∑

j=1

S·jYj(

∫ t

0fj(x, s)ds)


〈x(t)〉|−j

x(t)− 〈x(t)〉|−j


Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉



x(t) = x(0) +

r∑

j=1

S·jYj(

∫ t

0fj(x, s)ds)


〈x(t)〉|−j

x(t)− 〈x(t)〉|−j


Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉



x(t) = x(0) +

r∑

j=1

S·jYj(

∫ t

0fj(x, s)ds)


〈x(t)〉|−j

x(t)− 〈x(t)〉|−j


Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉



x(t) = x(0) +

r∑

j=1

S·jYj(

∫ t

0fj(x, s)ds)


〈x(t)〉|−j

x(t)− 〈x(t)〉|−j


Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉



x(t) = x(0) +

r∑

j=1

S·jYj(

∫ t

0fj(x, s)ds)


〈x(t)〉|−j

x(t)− 〈x(t)〉|−j


Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉



x(t) = x(0) +

r∑

j=1

S·jYj(

∫ t

0fj(x, s)ds)


〈x(t)〉|−j

x(t)− 〈x(t)〉|−j


Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉



x(t) = x(0) +

r∑

j=1

S·jYj(

∫ t

0fj(x, s)ds)


〈x(t)〉|−j

x(t)− 〈x(t)〉|−j


Σ(j)(t) ≡ 〈(x(t)− 〈x(t)〉|−j)(x(t)− 〈x(t)〉|−j)T〉


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)


Noise decomposition



Σ(t) = Σ(1)(t) + ... + Σ(r)(t)

dΣ


dΣ(j)



Noise decomposition



Σ(t) = Σ(1)(t) + ... + Σ(r)(t)

dΣ


dΣ(j)



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



∅ k+−→ x


dx = (k+ − k−x(t))dt +√


+√


Σ =12〈x〉︸︷︷︸

birth noise

+12〈x〉︸︷︷︸

death noise



∅ k+−→ x


dx = (k+ − k−x(t))dt +√


+√


Σ =12〈x〉︸︷︷︸

birth noise

+12〈x〉︸︷︷︸

death noise



∅ k+−→ x


dx = (k+ − k−x(t))dt +√


+√


Σ =12〈x〉︸︷︷︸

birth noise

+12〈x〉︸︷︷︸

death noise


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.



[Σ(r)

]nn

=12

[Σ]nn


[Σ(r)]nn =12〈xn〉




[Σ(r)

]nn

=12

[Σ]nn


[Σ(r)]nn =12〈xn〉



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

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



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



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



∅ 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




∅ 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




∅ 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




∅ 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



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.


Acknowledgement

Michael StumpfImperial College London

Jacek MiekiszUniversity of Warsaw

Barbel FinkenstadWarwick University

David RandWarwick University


Education

Calculus of biochemical noise