identification of noise sources in biochemical networks

Identification of noise sources in biochemicalnetworks

Michał Komorowski

Institute of Fundamental Technological ResearchPolish Academy of Sciences

19/06/14

Michał Komorowski 19/06/14

SensingTransmission

DecisionAdaptation


SensingTransmission

DecisionAdaptation

Nucleus

Cytoplasm


Nucleus

Cytoplasm


−σ

−σ

−σ+σ

+σ

+σ

Time

Nucleus

Cytoplasm

Outputdistribution

Out

put


−σ

−σ

−σ+σ

+σ

+σ

Time

Nucleus

Cytoplasm

20%

10%

20% 20%

10%

20%

Variance decomposition

Outputdistribution

Out

put


−σ

−σ

−σ+σ

+σ

+σ

Time

Nucleus

Cytoplasm

Variance decomposition

Outputdistribution

Out

put

0 2 4 6 8 10 15 20 30 40 50 600

2000

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18000

time (minutes)


−σ

−σ

−σ+σ

+σ

+σ

Time

Out

put


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


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 (Counts firing of j-th reaction)

How to calculate Σ(i)(t) = 〈(x(t)− 〈x(t)〉(i))2〉 ?

x(t) = φ(t) + ξ(t)

φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)

dξ = SOφF(φ)ξdt + S(

diag{√

F(φ)})

dW


Model equations

x(t) = x(0)

+

r∑j=1

S·jYj

(∫ t

0fj(x, s)ds

)





x(t) = φ(t) + ξ(t)

φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)


diag{√

F(φ)})

dW


Model equations

x(t) = x(0) +

r∑j=1

S·j

Yj

(∫ t

0fj(x, s)ds

)





x(t) = φ(t) + ξ(t)

φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)


diag{√

F(φ)})

dW


Model equations

x(t) = x(0) +

r∑j=1

S·jYj

(∫ t

0fj(x, s)ds

)





x(t) = φ(t) + ξ(t)

φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)


diag{√

F(φ)})

dW


Model equations

x(t) = x(0) +

r∑j=1

S·jYj

(∫ t

0fj(x, s)ds

)



Linear Noise Approximation


x(t) = φ(t) + ξ(t)

φ(t) = φ(0) +r∑

j=1

S·j(∫ t

0fj(φ, s)ds)


diag{√

F(φ)})

dW


Model equations

x(t) = x(0) +

r∑j=1

S·jYj

(∫ t

0fj(x, s)ds

)



Linear Noise Approximation


x(t) = φ(t) + ξ(t)

φ(t) = φ(0) +

r∑j=1

S·j(∫ t

0fj(φ, s)ds)


diag{√

F(φ)})

dW


In the linear noise approximationdΣ

dt= A(t)Σ + ΣA(t)T + D(t)

Σ(t) = Σ(1)(t) + ... + Σ(r)(t)

dΣ(j)

dt= A(t)Σ(j) + Σ(j)A(t)T + D(j)(t)


Arbitrary system

Komorowski M., Miekisz J., Stumpf M.P.H. Decomposing Noise in Biochemical Signalling Systems Highlights the Role of ProteinDegradation, Biophys J. 2013


Arbitrary system

No approximations!Only stationarity required.



Monomolecular reactions



Intuition



Intuition

- �xed production process



Intuition



Double feedback



Double feedback

0 10 20 30 40

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1.0

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Output Intensity

Prod

uctio

n an

d de

grad

atio

n ra

tes

Reno

rmal

ized

Pro

babi

lity

Den

sity

of O

utpu

t

−σ

−σ

−σ

+σ

+σ

+σ

Out

put I

nten

sity



Double feedback

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Output Intensity

Prod

uctio

n an

d de

grad

atio

n ra

tes

Reno

rmal

ized

Pro

babi

lity

Den

sity

of O

utpu

t

σ−

σ−

σ−

σ+

σ+

σ+

σ

Out

put I

nten

sity



StochDecompMatlab package:

sourceforge.net/p/stochdecomp/

Model definition (input)

Stoichiometry matrix Reaction rates Initial parameter values

SBM

L

Jetka et al. StochDecomp–Matlab package for noise decomposition in stochastic biochemical systems, Bioinformatics (2014)



StochDecompMatlab package:


Sym

bolic

com

puta

tions

Model equations

Model definition (input)

Stoichiometry matrix Reaction rates Initial parameter values

Deterministic state

Variance

Contributions

SBM

L




Linear cascades

for very generic signaling systems; we can even prove twogeneral propositions that assign surprisingly substantialcontributions to the overall output noise to the reactionthat describes the degradation of the output signal and these

results can be made mathematically precise and hold for theChemical Master Equation without the LNA (see the Exper-imental Methodology for proofs and details). Under suitablygeneral conditions (such as the single stable equilibrium

A

B C

D E

FIGURE 1 (A) Ten illustrative output trajectories representing stochastic dynamics of a hypothetical system. The observed intensity (e.g., experimentallymeasured fluorescence) exhibits stochastic dynamics, which results from each of the reactions present in the system. Equation 3 allows us to dissect thetotal variability. Decomposition describes how much of the observed variability is generated by each of the involved reactions. On the right-hand side is theprobability distribution of system outputs, the variance of which is considered in our decomposition. (B) Three-step open conversion system with rates fjfor reactions j ! 1,., 6 have the form f1 ! k"1, f2 ! k"2 x1, f3 ! k"3 x2, f4 ! k#4 x1, f5 ! k#5 x5, and f6 ! k#6 x6. (C) Variance contributions of thedifferent reactions in the three-step linear conversion system with parameters k"1 ! 50, k"2 ! 1, k"3 ! 1, k#4 ! 1, k#5 ! 1, and k#6 ! 1 or the slowconversion example, and k"1 ! 50, k"2 ! 10, k"3 ! 10, k#4 ! 1, k#5 ! 1, and k#6 ! 1 for the fast conversion case; all rates are per hour. (D) Three-stepopen conversion system with rates fj as in panel B. (E) Variance contributions of the different reactions in the three-step linear catalytic cascade withparameters k"1 ! 50, k"2 ! 0.1, k"3 ! 0.1, k#4 ! 1, k#5 ! 0.1, and k#6 ! 0.1, for slow catalysis, and k"1 ! 50, k"2 ! 10, k"3 ! 10, k#4 ! 1,k#5 ! 10, and k#6 ! 10 for fast reactions.

Biophysical Journal 104(8) 1783–1793

Decomposing Noise 1785



Gene expression

REDUCING THE CONTRIBUTION OFDEGRADATION NOISE

Our mathematical results and examination of the examplesabove demonstrate that the noise resulting from the degrada-tion of the output of a biochemical reaction system accountsfor a substantial fraction of the output variability. An effec-tive mechanism of noise suppression, therefore, should takethis observation into account. One possibility to controlnoise attributable to the degradation reaction is to controlits rate using a positive feedback loop. The consequencesof a controlled degradation had previously only been dis-cussed in the context of biochemical noise and comparedwith the negative feedback loop on production (21).

Here we propose the simple extension of a positive feed-back loop on the degradation reaction, taking into accountthat our propositions arise on the grounds of the equilibriumnoise balance between production and degradation. In anysystem at equilibrium the rates of the production and degra-dation reactions have to equal. The noise resulting fromeach of these reactions, as directly related to the rate, canbe attenuated only if both equilibrium rates are simulta-neously reduced. Therefore, combination of the positivefeedback loop (21) on degradation with the negative feed-back on production is more efficient than either of theseloops separately. This is not only because the control istighter but, importantly, also because intensity of these reac-tions at equilibrium is reduced.

For illustration and simplicity we consider a simple birth-and-death process, but the same mechanism is valid forgeneral systems considered in Proposition 1. Here mole-cules x arrive at rate k! and degrade at rate k"x. In theLNA this process can be expressed by the followingstochastic differential equation:

dx #!k! " k"x$t%

"dt !

#####k!

pdW1|$$$$${z$$$$$}

birth noise

!################k"hx$t%i

pdW2|$$$$$$$$$$${z$$$$$$$$$$$}

death noise

: (8)

The stationary distribution of this system is Poisson with themean hxi # k!=k": Therefore, in the stationary state, deathevents occur at rate k"hxi, which must be equal to the birthrate k!. The noise terms in the above equation are equal atstationarity, indicating that contributions of birth-and-deathreactions are equal. It is straightforward to verify that thedecomposition

S # 1

2hxi

|${z$}birth noise

! 1

2hxi

|${z$}death noise

(9)

holds indeed.Now suppose that the births and deaths rates are

denoted by f(x) and g(x), respectively, and the system isdescribed by

dx # $f $x%"g$x%%dt !############f $hxi%

pdW1|$$$$$$$$${z$$$$$$$$$}

birth noise

!############g$hxi%

pdW2|$$$$$$$$${z$$$$$$$$$}

death noise

: (10)

A E

B C D F

FIGURE 3 (A) Illustration of a protein expression and activation system with seven reactions; mRNA, protein, and activated protein are described in termsof their production, degradation, and potentially reversible (de-)activation through (de-)phosphorylation through kinases and phosphatases. (B) Variancedecomposition of the noise in the output (active protein) when activation is not reversible. (C) Output variance decomposition for slow dephosphorylation.(D) Output variance for decomposition for fast dephosphorylation. For panels A–C, we use the following parameters: k!1 # 40, k"1 # 2, k!2 # 2, andk"2 # 1; for (A) kphos # 0.5 and kdephos # 0; for (B) kphos # 0.5 and kdephos # 0.1; for (C) kphos # 0.5 and kdephos # 1. (E) Contributions to the varianceof the reactions involved in gene expression with fluctuating promoter states; here the transcription rate is modeled as k!1 # $Vy=H%=$1! y=H%, and y denotesthe stationary solution of dy # $b" gy%dt !

######2g

pdW. The following parameters were used: V # 100, H # 50, k!2 # 2k"1 # 0.44, and k"2 # 0.55. For fast

promoter fluctuations we used b# 50 and g# 1, and for slow promoter fluctuations we used b# 0.5 and g# 0.01. (F) Variance contributions of the reactionsinvolved in fluorescent protein maturation model for slowly (left) and quickly (right) maturing proteins (slow and fast proteins). The following parameterswere used: k!1 # 50, k"1 # 0.44, k!2 # 2, and k"2 # 0.55. For slow maturation (average maturation timez 5 h) we assumed folding and maturation rates tobe kf # 0.2 and km # 1.3, respectively. For fast maturation (average maturation time z 0.5 h) we set kf # 2.48 and km # 13.6. All rates are per hour.


1788 Komorowski et al.



Michaels-Menten kinetics

from the incoming and outgoing reactions. This simplyreflects that we can treat each molecular species as a poten-tial output, whence our statements from above apply.

Protein expression

The canonical example of a linear catalytic pathway, whichhas been widely explored in the context of noise decompo-sition (2,6,7,9–11), is gene expression. It can simply beviewed as the production of RNA (x1) from source (DNA)at rate k1

!, and production of protein (x2) in a catalytic reac-tion at rate k2x1, together with first-order degradation of bothspecies at rates k1

"x1 and k2"x2. Here we use this model to

demonstrate the applicability of our framework by revisitingearlier studies of Paszek (9), Paulsson (10), and Rausen-berger and Kollman (11), which reported noise contribu-tions arising at each level of expression (promoter statefluctuations, transcription, translation). Proposition 1 statesthat the part of the variance resulting from the proteindegradation equals half of the mean protein level,and Eq. 3 allows us to derive the complete decompositionof the protein, x2, variance

Sx2 #1

2hx2i

|!!{z!!}prot: degradation

! 1

2hx2i

|!!{z!!}translation

! 1

2

kphx2igr ! gp|!!!!!!{z!!!!!!}

mRNA degradation

! 1

2

kphx2igr ! gp|!!!!!!{z!!!!!!}

transcription

: (7)

In Fig. 3 A, we consider a slightly more general model,where we also consider activation of the protein throughphosphorylation and corresponding deactivation throughdephosphorylation; both kinase and phosphatase areassumed to be abundant and their respective activitiesassumed to be constant. Fig. 3, B–D, exemplifies how thenoise in the output (active protein) changes as dephosphor-ylation gains in importance. Increasing the rate of dephos-

phorylation slightly decreases the relative contributionmade to the output signal that results from transcription(R1), translation (R3), and mRNA and (inactive) proteindegradation (R2 and R4, respectively); these decreases arecompensated for by a marked increase to the system outputnoise resulting from the phosphorylation. This, too, is in linewith intuition suggesting that increasing later rates mustdecrease the contribution resulting from earlier reactions.

To illustrate applicability of this framework further, wealso investigate two extensions of the conventional proteinexpression model:

First, we assume that the promoter can fluctuate betweenon- and off-states (similarly to Paszek (9) and Rausenbergerand Kollman (11)) and calculate contributions for differenttimescales of these fluctuations, but focusing on proteinlevels per se (rather than just active protein).

Second, we assume that the protein is a fluorophore thatundergoes a two-step maturation process before it becomesvisible (folding and joint cyclization with oxidation).Fig. 3 E presents contributions for fast and slow promoterkinetics, showing that fast fluctuations are effectivelyfiltered out (contributing 10%) but remain a substantial con-tributor when they are slow (contributing 40%).

Variability in gene expression is often measured by meansof fluorescent proteins that undergo maturation beforebecoming visible for detection techniques; but the processof maturation (17) itself is subject to stochastic effects,and can thus contribute significantly to the observed vari-ability. We used typical parameters (17,18) for fast andslow maturing fluorescent proteins and found that matura-tion contributed 4 and 25%, respectively (Fig. 3 F) to theoverall variability; here our method allows for the rigorousquantification of the effect’s single reactions, comparedto previous analyses of steady-state statistics which wereonly able to consider the total noise levels (19,20).

FIGURE 2 Variance decomposition for the fourspecies of the Michaelis-Menten kinetics model.The following parameters were used: k0 # 0.1,k1 # 50, k2 # 1, kb # 20, and kd # 0.1. Numberof enzyme molecules was set to 50. All rates areper hour.


Decomposing Noise 1787



JAK-STAT pathway

JAK JAK JAKSTAT

Epo

p JAK p

p p

Nucleus

Cytoplasm

STAT

STATSTAT

p

p

STATSTAT

p

p

STAT

STAT

p

p

STAT

STAT

R1

R2

R3

R4-R12R13



JAK-STAT pathway

JAK JAK JAKSTAT

Epo

p JAK p

p p

Nucleus

Cytoplasm

STAT

STATSTAT

p

p

STATSTAT

p

p

STAT

STAT

p

p

STAT

STAT

R1

R2

R3

R4-R12R13

Data

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0.7

0.8

0.9

1

time (minutes)

Swameye I. et al. 2003



JAK-STAT pathway

JAK JAK JAKSTAT

Epo

p JAK p

p p

Nucleus

Cytoplasm

STAT

STATSTAT

p

p

STATSTAT

p

p

STAT

STAT

p

p

STAT

STAT

R1

R2

R3

R4-R12R13

Posterior distribution

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

1.5

2.0

2.5

3.0

3.5

4.0

Data

0 10 20 30 40 50 600

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0.8

0.9

1

time (minutes)

Swameye I. et al. 2003 Vanlier J. et al. 2012



JAK-STAT pathway

JAK JAK JAKSTAT

Epo

p JAK p

p p

Nucleus

Cytoplasm

STAT

STATSTAT

p

p

STATSTAT

p

p

STAT

STAT

p

p

STAT

STAT

R1

R2

R3

R4-R12R13

Posterior distribution

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

1.5

2.0

2.5

3.0

3.5

4.0

Data

0 10 20 30 40 50 600

0.1

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0.3

0.4

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0.9

1

time (minutes)

0 2 4 6 8 10 15 20 30 40 50 600

2000

4000

6000

8000

10000

12000

14000

16000

18000

time (minutes)

i=13

i=1i=2i=3i=4,...,12

Swameye I. et al. 2003 Vanlier J. et al. 2012



Implications

Degradation is a relevant component of noise

Contributions can be easily computed - StochDecmp

”Non-poissonian” rections

Extensions


Acknowledgement

Michael StumpfImperial College London

Tomasz Jetka

Agata Charzynska

Anna Gambin

Warsaw University

Jacek MiękiszWarsaw University

Acknowledgement

Tomasz Jetka

Tomasz Winarski

Phd students

Karol Nienałtowski

Szymon Majewski

Edyta Głów

MSc students

identification of noise sources in biochemical networks

Science

time output micha komorowski

death process micha

time minutes micha komorowski

xn x1 s1j

model equations xt

xn snj reaction rates

x0 r j

poisson birth