LiNA A Graphical Matlab Tool for Analyzing Intrinsic …LiNA A Graphical Matlab Tool for Analyzing Intrinsic Noise in Biochemical Reaction Networks Tutorials and Theoretical Background

LiNA � A Graphical Matlab Tool for

Analyzing Intrinsic Noise in

Biochemical Reaction Networks

Tutorials and Theoretical Background

Ronny Straube and Axel von Kamp

Analysis and Redesign of Biological Networks Group

Max Planck Institute for Dynamics of Complex Technical Systems

Sandtorstrasse 1, 39106 Magdeburg, Germany

August 19, 2013

Contents

1. Installation and License Information 4

1.1. Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. License Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4. Contact Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Tutorials 5

2.1. Noise Reduction Through Receptor Dimerization . . . . . . . . . . . . . . 52.1.1. Setting up the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2. Taking A First Look . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3. Volume Dependence of the Fluctuations . . . . . . . . . . . . . . . 82.1.4. Fano Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.5. Parameter Sets, Sessions and Figure Export . . . . . . . . . . . . . 10

2.2. Ultrasensitivity in the Goldbeter-Koshland Model . . . . . . . . . . . . . . 112.2.1. Analysis of Systems with Conservation Relations . . . . . . . . . . 122.2.2. Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3. Bistability in Covalent Modi�cation Cycles . . . . . . . . . . . . . . . . . . 152.3.1. Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2. Strategies for Dealing with Multiple Steady States . . . . . . . . . 16

3. Technical Remarks 19

3.1. Solving Eqs. (3.1) and (3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2. Rescan vs. All Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3. Data Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4. Output at MATLAB Console and vanKampen.out . . . . . . . . . . . . . 20

4. Theoretical Background 21

4.1. Reaction Networks and the Master Equation . . . . . . . . . . . . . . . . . 214.2. The Linear Noise Approximation . . . . . . . . . . . . . . . . . . . . . . . 224.3. Calculation of Stochastic Quantities . . . . . . . . . . . . . . . . . . . . . 244.4. Dealing with Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . 254.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5.1. Noise Reduction Through Receptor Dimerization . . . . . . . . . . 284.5.2. Receptor-Ligand Binding with Mass Conservation . . . . . . . . . . 32

A. Derivation of the Linear Noise Approximation 36

A.1. Transformation of the Master Equation . . . . . . . . . . . . . . . . . . . . 36

2

Contents

A.2. E�ective Equations for ~x (t) and Π(~ξ, t). . . . . . . . . . . . . . . . . . . 39

A.3. Equations for 〈ξi (t)〉 and σii′ (t) = 〈ξiξi′〉 . . . . . . . . . . . . . . . . . . . 40

3

1. Installation and License Information

1.1. Availability

LiNA can be downloaded from

http://www.mpi-magdeburg.mpg.de/projects/LiNA

1.2. Installation

Unzip the LiNA.zip archive to your HOME directory or to any other directory of yourchoice. This will create a directory called LiNA containing the program �les. Examplenetworks described in the Tutorial Chapter (→ Chapter 2) can be found in the Examplesfolder which is located in the LiNA directory.

1.3. License Information

LiNA is free software. It is published under the terms of the GNU General PublicLicense v3. A full copy of the license terms and conditions can be found in the �lenamed COPYING which is located in the LiNA directory.

1.4. Contact Information

Please send your comments, suggestions or bug reports to

[email protected]

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2. Tutorials

If you are familiar with the linear noise approximation you may want to have a lookat the following Tutorials which cover all of the functionality implemented in LiNA. Ifyou don't know the linear noise approximation you may �nd it useful to �rst read thebackground information in Chapter 4 and Appendix A before coming back.Each tutorial highlights di�erent aspects of LiNA's functionality. For example, in the

�rst tutorial one learns how to setup a model, how to de�ne compounded parametersand how to use the graphical user-interface for analyzing basic system properties. Thistutorial also covers data management tasks such as loading/saving sessions and exporting�gures. The second tutorial shows how LiNA handles systems with conservation relationsand how to analyze correlations. The last tutorial discusses the functionality providedto analyze systems with multiple steady states. This tutorial refers to LiNA v1.

2.1. Noise Reduction Through Receptor Dimerization

In the �rst tutorial we shall analyze how dimerization a�ects the steady state �uctuationsof receptor molecules. To this end, we consider the reaction system

∅ks�kd1

R (2.1)

R+Rk+

�k−

R2 (2.2)

R2kd2→ ∅ (2.3)

which represents an extension of the model considered by Hayot and Jayaprakash [5].In the absence of dimerization, synthesis and degradation of receptor monomers, asdescribed by Eq. (2.1), lead to Poissonian �uctuations where the average number ofmonomers is equal to the variance so that the Fano factor is 1. Hayot and Jayaprakashshowed that taking dimerization into account (Eq. 2.2), but neglecting dimer degrada-tion (Eq. 2.3), the steady state �uctuations for both, monomers and dimers, remainPoissonian within the linear noise approximation [5]. In Section 4.5.1 we re-analyzethe system considered by Hayot and Jayaprakash, but for a �nite lifetime of receptordimers (kd2 6= 0), which is shown to lead to sub-Poissonian �uctuations for monomersand dimers. Interestingly, these �uctuations can be minimized either for monomers orfor dimers depending on the parameter regime. In the following, we will use LiNA tovisualize some of the theoretical results obtained in Section 4.5.1.

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2. Tutorials

Figure 2.1.: Startup window of LiNA.

2.1.1. Setting up the Model

To start LiNA, startup MATLAB �rst. If necessary change to the LiNA directory (seeSection 1.2). Then, type startlina at the MATLAB prompt and press ENTER. Thiswill bring up the Equations window (see Fig. 2.1) which allows you to enter new reac-tions, specify initial parameter values and to de�ne new parameters through parametersubstitutions. You can also load an existing model using the Load... button. In theopening �le selection window change to the Examples folder and select the �le recep-tor_dimerization.mat after which the Equations window will look as in Fig. 2.2.

The Equation Editor

In the �rst and in the third reaction the digit `0' is used to denote unspeci�ed sources andsinks. Irreversible reactions are entered using a single arrow (->), reversible reactionsusing a double arrow (<->). New reactions can be added with the Add Reaction button.Further information on this topic can be obtained by pressing the ? button in the upperright corner of the Equations window. Reactions can be deleted using the delete button(red circle with the white cross) which is located to the left of each reaction �eld.For each reaction at least one parameter has to be speci�ed; two for reversible reactions.

The parameter in the left column refers to the forward reaction (→) whereas that in theright column refers to the backward reaction (←). Parameter names are prede�ned, butcan be changed (only alphanumeric characters).

A Note on Units

The unit of a parameter depends on the order of the reaction and on the chosen unitfor the species concentration. The default unit for the species concentration is µM =10−6mol/liter. In that case, the value of the parameter ks, which de�nes the rate withwhich monomers are synthesized (0→ R) , is expected to be entered in units of µM/timewhere `time' could be seconds, minutes or hours. The particular time unit is unimportant

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2. Tutorials

Figure 2.2.: Model setup for the reaction system in Eqs. (2.1) - (2.3).

since, in its current form, LiNA only calculates noise characteristics under steady stateconditions which do not depend on the speci�c time scale. However, it is importantthat all parameter values are given with respect to the same time scale. The �rstorder rate constants kd1 (kd1), kd2 (kd2) and k

− (km), which describe the degradation ofmonomers, dimers and the dissociation of the dimer, respectively, have the unit 1/timewhere the time unit has to be the same as that used for ks. Finally, the second-order rateconstant k+ (kp), which describes the association of two monomers into one dimer, isexpected to be entered in units of 1/(µM · time). If the species concentration is changed,e.g. from µM to nM , the values for zeroth and second order rate constants (ks and k

+),are re-interpreted accordingly, i.e. as nM/time for ks and 1/(nM · time) for k+. Thevalues of �rst order rate constants are not a�ected by such a change.

Why do I need to de�ne the reaction volume?

Calculation of the standard deviation and, hence, calculation of the coe�cient of variationdepends on the reaction volume (cf. Eq. 4.17). The default volume is set to V =10−15l = 1fl = 1µm3 which corresponds to the typical volume of a bacterial cell. Insuch a small volume a concentration of x = 10nM means that there are approximatelyn = V ·NA · x ≈ 6 particles in the system (NA = 6.022 · 1023 particles/mol). This is theregime where one can expect stochastic e�ects to become important.

Parameter Substitutions

In Subsection 4.5.1 we show that the steady states as well as the steady state �uctuationsof the reaction system in Eqs. (2.1) - (2.3) only depend on certain parameter combinations

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2. Tutorials

which can be chosen as (cf. Eq. 4.49)

Rs =kskd1

, α =kd1kd2

, β =kd2k−

, KD =k−

k+. (2.4)

In order to compare the results, generated by LiNA, with the theoretical results fromSubsection 4.5.1 it is, thus, helpful to be able to de�ne the same parameters in LiNA.This is the purpose of the Parameter Substitutions �eld shown in Figure 2.2. Usingthe expressions in Eq. (2.4) the parameters kd2, kd1, ks and k

+ have been replaced byRs (Rs), α (a), β (b) and KD (Kd) according to

kd2 = βk−, kd1 = αβk−, ks = Rsαβk−, k+ = k−/KD.

The biological meaning of these parameters is immediately evident: Rs represents thesteady state level of monomers in the absence of dimerization, α di�ers from 1 if monomersand dimers are degraded at di�erent rates, β compares the time scales for the dimerdegradation and dimer dissociation, and KD represents the dimer dissociation constant.

2.1.2. Taking A First Look

If the model setup is �nished the analysis can be started by pressing the Start button(Fig. 2.2). This will bring up two new windows, denoted as Control and Plots (Fig. 2.3),while the Equations windows becomes `frozen'. The �rst quantity to be plotted is alwaysthe Mean Values of all species as a function of the �rst entry in the parameter list ofthe Parameters �eld (A), Kd in this case. The parameter list contains the newly de�nedparameters (cf. Eq. 2.4) as well as k−(km) which has not been substituted. Note,however, that neither the steady states nor any of the stochastic quantities dependsexplicitly on k−. You can readily check this by choosing k− (km) from the Parameterpull-down list in the X-Axis:Variable Parameter �eld (B). Then you should see 2 straightlines parallel to the x-axis.

2.1.3. Volume Dependence of the Fluctuations

To see how the reaction volume a�ects the magnitude of steady state �uctuations dueto intrinsic reaction noise choose again Kd from the Parameter pull-down list in theX-Axis:Variable Parameter �eld (B) and select Mean with Errorbars from the pull-downlist in the Y-Axis �eld (C). This should add some errorbars to the steady state curves.The errorbars represent the standard deviation as de�ned in Section 4.3. Now, if thereaction volume is decreased the average number of particles also decreases and particle�uctuations become larger. Conversely, if the reaction volume is increased particle �uc-tuations become reduced. You can test this idea by changing the value of the reactionvolume: Select Vol. in the parameter list of the Parameters �eld (A) and change thevalue from 1e− 15 to 1e− 14 and, subsequently, to 1e− 16. Don't forget to press ApplyChanges for parameter changes to take e�ect.

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2. Tutorials

Figure 2.3.: Control window (left) and Plots window (right) of LiNA.

2.1.4. Fano Factor

If monomers and dimers are degraded at the same rate (α = 1) and if dimer degrada-tion occurs at a much slower rate than dimer dissociation (β � 1) the Fano factors ofmonomers and dimers can be approximated by (cf. Eqs. 4.55 and 4.56)

F1 =σs11xs1≈ 1− xs1/KD

(1 + 4xs1/KD)2

F2 =σs22xs2≈ 1− 1

4

(4xs1/KD)2

(1 + 4xs1/KD)2

where xs1 and xs2 are the steady state values of monomers (Eq. 4.47) and dimers(Eq. 4.48), respectively. Hence, �uctuations in both, monomers and dimers, are sub-Poissonian. However, whereas monomer �uctuations, again, become Poissonian (F1 ≈ 1)as xs1 � KD/4 dimer �uctuations are reduced and the Fano factor approaches F2 ≈ 3/4.To reproduce these results with LiNA choose Rs (Rs) from the Parameter pull-down

list in the X-Axis:Variable Parameter �eld (B), change the Range from 0.01 to 1000,change to log scale and choose Fano Factor from the pull-down list in the Y-Axis �eld(C). This should produce two curves as in Figure 2.4. Convince yourself that changingthe binding a�nity KD (Kd) merely shifts these curves to the left (if Kd is decreased)or to the right (if Kd is increased). Also, observe that increasing β while keeping α = 1�xed successively shifts the Fano factor curve for monomers below that for dimers, sothat monomer �uctuations are reduced. Now, keep β = 0.01 �xed and reduce α so thatα � 1. This reduces the Fano factor for monomers at intermediate values of Rs wheredimer �uctuations are still approximately Poissonian (region A in Fig. 2.5). At large

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2. Tutorials

Figure 2.4.: Fano factors for monomers (R) and dimers (R2) as a function of Rs = ks/kd1for KD = 1µM , α = 1 and β = 0.01.

values of Rs the situation is reversed (region B in Fig. 2.5). Hence, if dimers are degradedat a much higher rate than monomers it depends on the value of Rs whether �uctuationsin monomers or dimers are reduced.

2.1.5. Parameter Sets, Sessions and Figure Export

As you keep playing around with di�erent parameter combinations LiNA will add acorresponding Parameter set for each distinct parameter combination. The number ofavailable parameter sets is shown in round brackets in the Parameters �eld. To choosea particular parameter set, simply change the number in the Parameter Set: �eld andpress Enter. When you change the variables along the x-axis or the y-axis or if youchange parameters LiNA adds a new �gure to the plot list which can be accessed fromthe pull-down menu in the Plots window (D in Fig. 2.3).The plots are labeled according to the format

parameter on the x-axis: quantity on the y-axis {parameter set}.

For example, in Fig. 2.5 the plot shows the Fano factor as a function of Rs usingparameter set 3. Note that parameter sets cannot be deleted. You can, however, deletea plot by pressing the Delete button next to the pull-down menu in the Plots window.If you also want to delete the calculations associated with a plot you have to use thedelete button (red circle with a white cross) next to the Parameter pull-down list in theX-Axis: Variable Parameter �eld (B in Fig. 2.3).

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2. Tutorials

Figure 2.5.: Fano factors for monomers (R) and dimers (R2) as a function of Rs = ks/kd1for KD = 1µM , α = 0.001 and β = 0.01.

If you wish to continue the analysis at a later time you can save the current sessionincluding parameter sets and plots using the save button (disk symbol right above theY-Axis �eld) in the Control window. Saved sessions can be loaded using the Resume

Session ... button from the startup window of LiNA (Fig. 2.1).If you want to save a �gure for further processing in MATLAB you can either directly

save the plot as a MATLAB �gure �le (using the disk symbol in the Plots window) oryou copy the contents of the current plot (using the copy symbol labeled by E in Fig.2.3) which creates a new MATLAB �gure so that you can add comments, change fontsor convert the �gure to another format before saving. Note that the plots in a copied�gure will not receive any further updates as you continue with your analysis.

2.2. Ultrasensitivity in the Goldbeter-Koshland Model

The Goldbeter-Koshland model is a classical model for the description of covalent modi-�cation systems [3] where a substrate molecule is interconverted between an unmodi�ed(S) and a modi�ed (S∗) form by a pair of opposing converter enzymes. In the case ofphosphorylation / dephosphorylation cycles the converter enzymes are called kinase (K)and phosphatase (P ). In the simplest case the enzyme-catalyzed reactions are modeledas irreversible Michaelis-Menten type reactions of the form

S +Kk+1�k−1

S-Kk1→ S∗ +K, S∗ + P

k+2�k−2

S∗-Pk2→ S + P. (2.5)

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2. Tutorials

Figure 2.6.: Model setup for the reaction system in Eq. 2.5.

Here, we have assumed that the reactions occur under in vitro conditions so thatsynthesis and degradation of substrate and converter enzymes can be neglected. Hence,there are three conservation relations

[S] + [S∗] + [S-K] + [S∗-P ] = ST

[K] + [S-K] = KT

[P ] + [S∗-P ] = PT

corresponding to the conservation of the total amounts of substrate (ST ), kinase (KT )and phosphatase (PT ).If the total substrate concentration is much larger than that of either converter enzyme

(ST � max (KT , PT )) one can describe the dynamics of the modi�ed form of the substrateby the e�ective equation [3]

d [S∗]

dt≈ k1

ST − [S∗]

K1 + ST − [S∗]− k2

[S∗]

K2 + [S∗]

whereK1 = (k1+k−1 )/k+1 andK2 = (k2+k−2 )/k+2 denote the Michaelis-Menten constantsof the kinase and the phosphatase, respectively. A hallmark of the Goldbeter-Koshlandmodel is that, under steady state conditions d [S∗] /dt = 0, it can generate highly sig-moidal response curves if both converter enzymes operate in saturation (Fig. 2.7), i.e. ifmax(K1,K2)� ST . This phenomenon is called zero-order ultrasensitivity.

2.2.1. Analysis of Systems with Conservation Relations

To setup the model start LiNA from the MATLAB prompt with startlina and enterreactions and parameters as shown in Fig. 2.6. Alternatively, you may load the �le Gold-

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2. Tutorials

beter_Koshland_model.mat from the Examples folder. Note that the `on' rate constantsk+1 and k+2 have been replaced by the associated Michaelis-Menten constants accordingto k+i = (ki + k−i )/Ki (i = 1, 2) where Ki ↔ Kmi. After starting the calculation (pressStart) you will see three new parameters (C_K, C_Sp and C_P) in the parameter listof the Parameters �eld (Fig. 2.7). LiNA automatically adds a new parameter for eachconservation relation it detects in the network. Hence, C_K, C_Sp and C_P corre-spond to the total concentrations of kinase, substrate and phosphatase. LiNA displaysthe corresponding conservation relation in the X-Axis: Variable Parameter �eld of theControl window once you select one of these parameters from the Parameter list.As networks become larger the number of species also increases. By default, LiNA

plots the mean values of all species in the Plots window after startup, which can beconfusing. Species can be selected or deselected by pressing the corresponding speciesbutton in the Variables �eld of the Control window. For large systems it can be fasterto deselect all species �rst and then select only those species that are of interest: Use theright mouse button while holding the mouse pointer over the Variables �eld name.To generate the ultrasensitive response curves shown in Fig. 2.7, choose C_K from the

Parameter list in the X-Axis:Variable Parameter �eld and change the value of C_Sp from1 to 10 so that ST � PT . Note that the condition max (K1,K2)� ST is already ful�lled.Now, adjust the upper boundary of the Range to 2 and increase the number of Intervalsfrom 10 to 40. Observe that both, S (S) and Sp (S∗), exhibit a sharp transition nearC_K=1 (KT = 1). This suggests that the steady state �uctuations of these two speciesare large near C_K=1. Convince yourself that this is true: Speci�cally, observe that theFano factors of S and Sp change by a factor of more than 20 near C_K=1 (Choose FanoFactor from the pull-down list in the Y-Axis �eld of the Control window). In contrast,the coe�cient of variation, which measures ratio between the standard deviation and themean value, changes much less dramatically. (Choose Coe�cient of Variation from thepull-down list in the Y-Axis �eld of the Control window)

2.2.2. Correlation Analysis

As systems become larger it can be interesting to study correlations among the species.As Figure 2.8 shows the strength of correlations changes as parameters are varied. Thereare two ways to start the correlation analysis. First, you can select Correlation from thepull-down list of the Y-Axis �eld in the Control window. By default, this will only plotthe correlation coe�cient among the �rst two species of the Variables �eld. Alternatively,you can press the Update Plot button in the Correlations �eld of the Control windowwhich has the same e�ect.To add new species pairs simply choose the corresponding species from the pull-down

lists in the Correlations �eld and press Add. You may also select All species or all Selectedspecies at once and press Add. For example, the combination Species/Selected will addall distinct pairs between Species and the selected species in the Variables �eld of theControl window. Similarly, the combination All/All would add all n(n − 1)/2 distinctspecies pairs (where n is the total number of species in the system) to the list in theCorrelations �eld. Species pairs in the Correlations �eld can be highlighted by clicking

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2. Tutorials

Figure 2.7.: Ultrasensitivity in the Goldbeter-Koshland model for K1 = K2 = 0.1µM ,k1 = k2 = 1/s, ST (C_Sp) = 10µM and PT (C_P) = 1µM . Note that thecurves for SK and SpP overlap.

on them (press the Shift or Control key for multiple selection). Highlighted species areremoved via a context menu that opens upon a right-click with the mouse button withinthe Correlations �eld. This method is especially useful if you want to remove a smallnumber of species pairs. However, imagine you have all n(n − 1)/2 species pairs in thelist and you want to remove only those that contain a particular species, say X. In thatcase it might be faster to choose the combination X /All from the pull-down list andpress the Remove button.Despite the fact that the existence of a correlation between species A and B, in general,

does not imply a sense of interaction between them, the analysis of correlations can stillprovide some hints for the functionality of a network. To this end, delete the S/K pairfrom the Correlations list in the Control window and add the pairs S/Sp, K/Sp andP/Sp and press Update Plot. As you can see the modi�ed and the unmodi�ed forms ofthe substrate are always anti-correlated (Fig. 2.8) which is a direct consequence of theapproximate conservation relation for the substrate ST ≈ [S] + [S∗], which holds undersubtrate excess. As the total substrate concentration is reduced this anti-correlationbecomes weaker. (Try reducing C_Sp!) The almost perfect correlation between S∗ (Sp)and K beyond the transition point C_K=1 means that essentially all of the modi�edsubstrate is generated through the reaction S-K → S∗ + K in the parameter regionC_K>1 and not through the dissociation of the S∗-P complex - in agreement with thefact that the correlation coe�cient for S∗ (Sp) and P is essentially zero for C_K>1.

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2. Tutorials

Figure 2.8.: Correlation coe�cients for selected species pairs in the Goldbeter-Koshlandmodel (Eq. 2.5). Parameter values are the same as those used in Fig. 2.7.

2.3. Bistability in Covalent Modi�cation Cycles

Here, we consider an extension of the Goldbeter-Koshland model which has been recentlyanalyzed in the context of bistability and hysteresis [8]. The elementary reactions read

S +KLk+1�k−1

S-KLk1→ S∗ +KL (2.6)

S + Pk+2�k−2

S∗-Pk2→ S + P

K + Lk+3�k−3

KL

P + Lk+4�k−4

PL

where L represents an allosteric e�ector that reciprocally a�ects the activities of theconverter enzymes. Speci�cally, binding of L is assumed to activate the kinase and toinactivate the phosphatase.It can be shown that the steady states of this system depend on the Michaelis-Menten

constants of the kinase (K1 = (k1 + k−1 )/k+1 ) and the phosphatase (K2 = (k2 + k−2 )/k+2 ),on the dissociation constants of the enzyme-e�ector complexes KL (KD3 = k−3 /k

+3 )

and PL (KD4 = k−4 /k+4 ), on the catalytic rate constants of the kinase (k1) and the

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2. Tutorials

Abbildung 2.9.: Steady state response curve for the extended Goldbeter-Koshland modelin Eq. (2.6). In the region between the two saddle-node bifurcations(SN) two stable steady states (solid lines) coexist with one unstablesteady state (dashed line). Parameter values: K1 = 1µM , K2 = 0.01µM ,KD3 = 10µM , KD4 = 0.01µM , k1 = 10/s, k2 = 0.1/s, ST = 3µM ,KT = PT = 1µM .

phosphatase (k2) as well as on the total concentrations of substrate (ST ) and converterenzymes (KT , PT ). If certain conditions among these parameters are met this systemcan exhibit bistability for a range of e�ector concentrations (Fig. 2.9).

2.3.1. Model Setup

Load the bistable_Goldbeter_Koshland_model.mat �le from the Examples folder. Notethat, in the Parameter Substitutions �eld, the `on' rate constants k+i (i = 1, . . . , 4) havebeen replaced by Michaelis-Menten (K1, K2) and dissociation constants (KD3, KD4)according to

k+i =ki + k−iKi

, for i = 1, 2 and k+i =k−iKDi

, for i = 3, 4.

After starting the calculations you will �nd 4 new parameters in the Parameters list of theControl window which correspond, respectively, to the total concentration of substrate(C_Sp), converter enzymes (C_K and C_P) and allosteric e�ector (C_L). Convinceyourself that the steady states do not depend on the backward rate constants k−i (kim).

2.3.2. Strategies for Dealing with Multiple Steady States

LiNA cannot generate bifurcation diagrams, as in Fig. 2.9, directly which would requirecontinuation methods [6]. Instead, LiNA uses a simple strategy to �nd the stable branchesof the steady state curve, either by using di�erent starting points in parameter space orby trying to numerically �nd all solutions of the steady state equations.

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2. Tutorials

First Strategy

To generate the stable branches of the steady state curve shown in Fig. 2.9, choose C_Lfrom the Parameter list in the X-Axis:Variable Parameter �eld, change the values of k1and C_Sp from 1 to 10 and 3, respectively, adjust the upper boundary of the Range to0.5, increase the number of Intervals from 10 to 40 and deselect all species except for Sp(S∗). This should result in a plot as in Fig. 2.10.Apparently, we have obtained the upper stable branch of the steady state curve in

Fig. 2.9 as well as the part of the lower branch that is to the left of the saddle-nodebifurcation (SN). The �rst strategy to obtain the lower stable branch is to generate thesteady state curve in the opposite direction. This can be done using the Rescan buttonsin the Multiple Steady States �eld of the Control window (A, Fig. 2.10). By repeatedlyusing the left and right arrow you alternatively obtain either the lower or the upper partof the steady state branch. (Try it!) If you wish to keep both branches press the copybutton right next to Solution branch: 1 in the Multiple Steady States �eld. This willcreate a copy of the current solution branch. Then, use again the Rescan buttons togenerate the complementary branch. Now you can access both branches independentlyeither through the Solution branch pull-down list in the Multiple Steady States �eld orthrough the pull-down list in the Plots window (B, Fig. 2.10). In the latter case, thebranch number is indicated in square brackets.

Second Strategy

If a system has more than two stable steady states it might be di�cult, if not impossible,to �nd all stable branches with the �rst strategy. In that case you may press the All

Steady States button in the Multiple Steady States �eld of the Control window. Thisfunction tries to �nd all steady state curves numerically. It checks for local stability andreturns the stable branches which should then appear in the Solution branch pull-downlist. For the current example, All Steady States yields the same result as the strategyusing the Rescan buttons (Try it!). In general, it seems reasonable to combine bothstrategies. The method using the Rescan buttons is fast and gives a �rst indicationwhether a system admits multiple stable steady states in a certain parameter range. AllSteady States can be used to test whether there are further stable branches which mayexist, for example, inside the region between two saddle-node bifurcations.Having found the stable branches of the steady state set one may plot the Fano factor

or any of the other quantities from the pull-down list of the Y-axis �eld along thesebranches similar as in the case of a single steady state curve. Note, however, that thepredictions, based on the LNA, may become worse close to the saddle-node bifurcationsas the LNA fails at such bifurcation points.

17

2. Tutorials

Figure 2.10.: Stable branches of the steady state curve in Fig. 2.9. Parameter values arethe same as in the caption of Fig. 2.9.

18

3. Technical Remarks

To generate parameter scans for graphical visualization LiNA solves the steady stateequations that result from the linear noise approximation of the master equation (cf.Eqs. 4.8 and 4.14)

d~x

dt= N · ~f0 (~x, ~p) = 0 (3.1)

dσ

dt= K (~x) · σ + σ ·KT (~x) + D (~x) = 0. (3.2)

For a given network topology (encoded by the stoichiometric matrixN) and a given set ofkinetic parameters (~p) Eq. (3.1) represents k, in general nonlinear, algebraic equations forthe unknown steady state concentrations of the k species ~xs = (xs1, . . . , x

sk). In contrast,

Eq. (3.2) results in a set of linear algebraic equations for the k (k + 1) /2 independentcomponents of the variance-covariance matrix σ. Hence, for a given set of parametersLiNA has to solve k + k (k + 1) /2 algebraic equations. From the values of ~xs and σ allother quantities of interest can be derived (cf. Section 4.3).Although the steady states are calculated numerically, at the beginning of the analysis

LiNA sets up Eqs. (3.1) and (3.2) symbolically in order to apply parameter substitutionsif desired. All symbolic calculations are performed with MuPAD which is available assymbolic toolbox for MATLAB since release 2007b. In order to make use of the symbolicresults they are converted into character strings in a format which can be processedby the MATLAB parser. For actual computations, parameters are substituted by theirvalues where necessary and the resulting strings are parsed into anonymous MATLABfunctions which then can be used as inputs for the numerical methods. In this mannerexact analytical results are used as far as possible and the numerical calculations canstill be performed e�ciently with the parsed functions.

3.1. Solving Eqs. (3.1) and (3.2)

The primary method for calculating steady states is the MATLAB root solver fsolve.Only non-negative steady states that ful�ll the conservation relations (when present)and are locally stable are accepted as valid. In case the root solver returns an invalidsteady state or fails to �nd a solution the ODE system in Eq. (3.1) is numericallyintegrated for a given amount of time and the resulting end point taken as a new startingpoint for the root solver. This procedure is iterated up to three times with the integrationtime increasing in each iteration. If an integration occurs a message is displayed at theMATLAB console. In order to improve the e�ciency, a linear interpolation of alreadyknown steady states is kept from which start points for the calculation of further steady

19

3. Technical Remarks

states are derived. To solve Eq. (3.2) the matricesK (~xs) andD (~xs) are evaluated at thesteady state solution ~xs found by solving Eq. (3.1). The equation is then transformedinto the standard form A · ~σ = ~b which can be readily solved using standard MATLABfunctions. The vector ~σ contains the independent components of the variance-covariancematrix. For example, in the case of a reaction network with 2 species it would be givenby ~σ = (σ11, σ22, σ12)

T .

3.2. Rescan vs. All Steady States

In case Eq. (3.1) admits multiple stable steady states one can use the functions Rescanor All Steady States from the Control window to �nd them. The Rescan function usesthe previous steady state as a starting point for calculating the next steady state alongthe parameter curve (in scan direction). In contrast, the function All Steady States triesto �nd all stable steady states in given parameter range by calling the MuPAD functionnumeric::solve. Although this is convenient in case of bifurcations it is also a quite time-consuming procedure. Note that neither rescan nor calculation of all steady states willintroduce any new support points along the parameter curve but rather operates on thosepoints that are visible in the current plot window.

3.3. Data Management

The primary results of steady states computations, i.e. the mean values and variances, arepermanently kept by LiNA unless they are explicitly cleared. They are stored separatelyfor each combination of parameter set and variable parameter that has been selected atany time. Covariances, which are needed for calculating correlations, are calculated forall pairs that have been displayed in a correlation plot for a given parameter set/variabelparameter combination at any time and will be kept from that point onwards.

3.4. Output at MATLAB Console and vanKampen.out

After starting the analysis by pressing the Start button in the Equation editor (cf.Fig. 2.2) a message is displayed at the console indicating whether or not parametersubstitution (if intended) has been successful and whether conservation relations havebeen detected. The console also displays the result of parameter scans each time aparameter or a parameter range is changed. In case something goes wrong during thecomputation an error message is displayed at the console which might help to tracepossible problems in the computation.For the record, the results of the symbolic computations are written to the �le vanKam-

pen.out each time a new analysis is started. Speci�cally, these are the stoichiometricmatrix N, the Jacobian K, the di�usion matrix D as well as the matrix A and thevector ~b which are used to solve Eq. (3.2). The �le also displays the right-hand side ofthe ODE system in Eq. (3.1) in symbolic form, lists the independent species togetherwith the reduced stoichiometric matrix NR and conservation relations (if present).

20

4. Theoretical Background

In this Chapter we provide the necessary theoretical background to appreciate the func-tionality implemented in LiNA. To make the presentation as self-contained as possible wegive a derivation of the linear noise approximation of the master equation for biochemicalreaction networks in Appendix A. We also describe how to deal with mass-conservationrelations. Speci�c examples are provided to illustrate the general method.

4.1. Reaction Networks and the Master Equation

We consider a volume V where r chemical reactions take place between k species. Sucha stochiometric reaction network can be written in the form

ν−1jX1 + . . .+ ν−kjXkWj→ ν+1jX1 + . . .+ ν+kjXk, j = 1, . . . , r (4.1)

where the index j numbers the reactions. From the stoichiometric coe�cients ν±ij ∈{0, 1, 2, . . .} one can construct the stoichiometric matrix N which has dimension k × r.Its components, which are de�ned by

Nij := ν+ij − ν−ij , i = 1, . . . , k, j = 1, . . . , r , (4.2)

indicate how many molecules of speciesXi are produced (Nij > 0) or consumed (Nij < 0)in the jth reaction.Let ni denote the number of molecules of species Xi. Then the state of the reaction

system is speci�ed by the probability P (ni, t) to have ni molecules of species Xi attime t in the system. In a Markovian description of the reaction network (Eq. 4.1) thedynamics of P (ni, t) is determined by transition rates Wj (ni +Nij |ni) which denote theprobability per unit time for a change of the number of molecules of species Xi by anamount Nij as a result of reaction j. For given transition rates the temporal evolution ofP (ni, t) is determined by the chemical master equation [9]

dP (~n, t)

dt=

r∑j=1

[Wj

(~n|~n− ~Nj

)P(~n− ~Nj , t

)−Wj

(~n+ ~Nj |~n

)P (~n, t)

](4.3)

where ~n = (n1, . . . , nk)T is a column vector whose components denote the molecule

numbers ni of species Xi and ~Nj = (N1j , . . . , Nkj)T represent a set of vectors that are

constructed from the jth column of the stoichiometric matrix as de�ned in Eq. (4.2).Despite the fact that Eq. (4.3) is linear in P (~n, t) the transition rates Wj often exhibit anonlinear dependence on ~n so that exact solutions of this equation are rare. As a result,one either relies on numerical simulations [2] or on approximation methods.

21


One such approximation method is the linear noise approximation (LNA) which isbased on van Kampen's system size expansion of the master equation [9]. To this end,one assumes that the molecule number vector can be decomposed as

ni = Vmxi + V12m ξi, i = 1, . . . , k (4.4)

where

xi :=〈ni〉Vm

(4.5)

represents the average concentration of species Xi and

Vm ≡ V ·NA

(NA ≈ 6 · 1023particles/mol

)(4.6)

denotes the molar volume. Here, we have followed standard biochemical conventionsaccording to which species concentrations are measured in units of mol/liter.The k quantities ξi in Eq. (4.4) represent the new stochastic variables describing the

deviations from the deterministic behavior. Hence, the idea of the decomposition in Eq.(4.4) is to separate the average dynamics, described by xi, from the �uctuations ξi whichare described by a new probability density Π (ξi, t).

4.2. The Linear Noise Approximation

The LNA is obtained by inserting the decomposition in Eq. (4.4) into the master equa-

tion (Eq. 4.3) and expanding the resulting equation in powers of V− 1

2m . The rationale

behind this procedure is that one expects that, in the limit of large system size V →∞,�uctuations become negligible and one obtains an e�ective equation for the average con-centration ~x. To perform the expansion one has to make an assumption about how the

transition rates Wj

(~n+ ~Nj |~n

)≡ Wj (~n) in Eq. (4.3) depend on the system size. By

looking at speci�c examples (see Section 4.5) it turns out that most cases of practicalinterest are covered by the Ansatz (cf. Ref. [9])

Wj (~n) = Vmf0,j

(~n

Vm

)+ f1,j

(~n

Vm

)+

1

Vmf2,j

(~n

Vm

)+ . . .

=∞∑l=0

V 1−lm fl,j

(~x+ V

− 12

m~ξ

)(4.7)

which ensures that, in the limit V → ∞, the functions fl,j only depend on the averageconcentration ~x and contributions from terms with l ≥ 2 become negligible for V →∞.Also, the assumption that the leading order term is proportional to V re�ects the factthat the probability for a particular transition increases with the reaction volume if theaverage concentration is kept constant. Later, we shall see that, in the LNA, the dynamicsof ~x and ~ξ in Eq. (4.4) is completely determined by the leading order term in Eq. (4.7).

22


For the average concentration ~x (t) the expansion of the master equation yields a systemof nonlinear ordinary di�erential equations (ODEs) given by (cf. Section A.2)

dxidt

=

r∑j=1

Nijf0,j (~x) , i = 1, . . . , k (4.8)

whereas the density of the �uctuations around the average is described by the linearpartial di�erential equation

∂Π(~ξ, t)

∂t=

k∑i,i′=1

−Kii′ (~x)∂[ξi′Π

(~ξ, t)]

∂ξi+

1

2Dii′ (~x)

∂2Π(~ξ, t)

∂ξi∂ξi′

, (4.9)

which is called Fokker-Planck equation. The drift matrix K and the di�usion matrix Din Eq. (4.9) are de�ned through

Kii′ (~x) :=r∑j=1

Nij∂f0,j (~x)

∂xi′and Dii′ (~x) :=

r∑j=1

f0,j (~x)NijNi′j . (4.10)

The nomenclature `linear noise approximation', which is used to denote Eq. (4.9), resultsfrom the fact that the coe�cients in front of the derivative terms in the Fokker-Planckequation are, at most, linear in ~ξ. From the de�nitions, given in Eq. (4.10), it is clearthat K represents the Jacobian matrix associated with the reaction system in Eq. (4.8).In general, the coe�cient matrices K (~x) and D (~x) depend implicitely on time throughtheir dependence on the average concentration ~x(t). Hence, to perform the LNA one�rst has to solve the ODE system in Eq. (4.8). Of course, in most practical cases thiscan only be done numerically. In a second step, one solves the Fokker-Planck equation(4.9) where the mean-�eld solution enters through the coe�cient matrices K (~x) andD (~x). However, since the Fokker-Planck equation is linear with respect to ~ξ its solutionis always a multivariate Gaussian distribution given by

Π(~ξ, t)

=1√

(2π)k detσ (t)exp

− k∑i,i′=1

(ξi − 〈ξi (t)〉)σ−1ii′ (t) (ξi′ − 〈ξi′ (t)〉)2

(4.11)

where σ−1ii′ are the components of the inverse variance-covariance matrix

σii′ = 〈ξiξi′〉 , i, i′ = 1, . . . , k (4.12)

and detσ 6= 0 denotes its determinant. Using the Fokker-Planck equation in Eq. (4.9)

it is straightforward to show that ~〈ξ〉 and σ are determined by the linear ODE systems(cf. Section A.3)

d ~〈ξ〉dt

= K · ~〈ξ〉 (4.13)

dσ

dt= K · σ + σ ·KT + D (4.14)

23


i.e. the average �uctuations ~〈ξ〉 obey the same equation as small deviations from a steadystate of the mean-�eld equations (Eq. 4.8). This means that the LNA is only applicablenear an asymptotically stable steady state where all eigenvalues of K have a negativereal part and the average �uctuations decay to zero in time. If the mean-�eld equations(Eq. 4.8) exhibit multiple stable steady states one can perform the LNA near each ofthe stable branches away from the bifurcation points.The variance-covariance matrix, as de�ned in Eq. (4.12), has the same units as the

average concentration ~x (cf. Eq. 4.4), i.e. mol/liter in our case. However, if oneis interested in particle number �uctuations rather than concentration �uctuations onehas to transform the Gaussian distribution in Eq. (4.11) back to original variables.Speci�cally, from Eq. (4.4) the following relations are apparent

ξi =ni − Vmxi

V12m

ξi − 〈ξi〉 =ni − Vmxi − V

12m 〈ξi〉

V12m

≡ ni − 〈ni〉

V12m

d~ξ = V− k

2m d~n

so that the Gaussian distribution in Eq. (4.11) can be rewritten as

P (~n, t) =1√

(2π)k detC (t)exp

− k∑i,i′=1

(ni − 〈ni (t)〉)C−1ii′ (t) (ni′ − 〈ni′ (t)〉)2

where

Cii′ = Vmσii′ (4.15)

denotes the variance-covariance matrix with respect to particle numbers. The dynamicsof C follows from that of σ by multiplying Eq. (4.14) by Vm which results in

dC

dt= K ·C + C ·KT + VmD .

4.3. Calculation of Stochastic Quantities

From the variance-covariance matrix σ (Eq. 4.14) and the mean-�eld solution ~x (4.8) onecan readily calculate several stochastic quantities of interest such as the Fano factor Fiand the coe�cient of variation CVi of the species Xi as well as the correlation coe�cientrii′ between species Xi and Xi′ . Using Eqs. (4.4) and (4.15) these quantities can beexpressed as

Fi =Cii〈ni〉

=σiixi, i = 1, . . . , k (4.16)

CVi =

√Cii〈ni〉

=

√σii√Vmxi

, i = 1, . . . , k (4.17)

rii′ =σii′√

σii√σi′i′

, i, i′ = 1, . . . , k . (4.18)

24


This shows that, within the LNA, the Fano factor and the correlation coe�cients are

independent of the reaction volume V . In particular, since ξi ∼ V− 1

2m the components of

the variance-covariance matrix σii′ = 〈ξiξi′〉 have the same unit as the mean concentra-tions xi. Also, from Eq. (4.17) we see that the standard deviation with respect to themean concentration is given by

√σii/Vm which has the same unit as xi. In contrast,

the coe�cient of variation depends inversely on the molar volume and, hence, the CVvanishes as the reaction volume V increases.

4.4. Dealing with Mass Conservation

Species that are not actively synthesized or degraded obey mass conservation relationswhich are indicated by linear dependencies between some rows of the stoichiometricmatrix N. This may happen, for example, if a transcription factor binds to speci�ctarget promoters on the DNA in which case the promoters can exist in two states: freeor occupied by a transcription factor, but the total amount of promoters is constant intime, at least over the time scale of the cell cycle. Another important example, wheremass conservation must be taken into account, is for the modeling of enzyme-catalyzedreactions under in vitro conditions. Here, substrates, enzymes and cofactors are typicallysupplied in �xed amounts so that their total concentrations remain constant in time.In general, there are as many linearly dependent rows of N as there are conserved

species in the system. Mathematically, mass conservation relations are associated withthe left null space of N which is de�ned by

S ·N = 0 . (4.19)

The dimension of S is (k − k′)×k where k′ is the number of linearly independent rows ofN, i.e. S has a many rows as there are conserved species in the system, and the numberof columns equals the number of species. In terms of S one can de�ne the vector ofconserved species ~s through

~s = S · ~x . (4.20)

Using the mean-�eld equations in Eq. (4.8) it is straightforward to see that ~s is conservedin time, i.e.

d~s

dt= S · d~x

dt= S ·N · ~f0 (~x) = 0 , (4.21)

which means that~s (t) = S · ~x (t) ≡ ~s0 ∀t (4.22)

where ~s0 is a constant vector whose components denote the total (molar) concentrationsof the conserved species. Writing Eq. (4.22) in terms of particle numbers, rather thanconcentrations, yields

S · ~n (t) = ~n0 (4.23)

where ~n0 = Vm~s0 now denotes total particle numbers of the conserved species.

25


Remark

For ~s0 (or ~n0) to be interpretable as a vector of total concentrations (or particle numbers)it is necessary that all elements of S are positive. However, solving Eq. (4.19) generallyyields a matrix S which contains both positive and negative entries. Hence, beforede�ning ~s through Eq. (4.20), one �rst has to �nd a positive basis of the left null spaceof N, e.g. through appropriate linear combinations of the rows of S.

The conservation relations in Eq. (4.23) are k − k′ linear equations for the k particlenumbers of species Xi (i = 1, . . . , k), which can be used to partition the Xi into k′

independent species XR,i (i = 1, . . . , k′) and k− k′ dependent species Yα (α = 1, . . . , k−k′). Let's denote by nR,i (xR,i) and mα (yα) the particle numbers (concentrations)associated with species XR,i and Yα, respectively. Then, the repartioned quantities canbe obtained from the original quantities according to

~n′ ≡(~nR~m

)= P · ~n and ~x′ ≡

(~xR~y

)= P · ~x (4.24)

where P is an appropriate permutation matrix. Using these relations in conjunctionwith Eq. (4.4) one can also decompose the �uctuations into those associated with theindependent species (~ξR) and those associated with the dependent species (~η) as

~ξ′ =

(~ξR~η

)= P · ~ξ (4.25)

so that particle number vectors of the dependent and the independent species can bewritten in the form

~nR = Vm~xR + V12m~ξR (4.26)

~m = Vm~y + V12m~η .

In the next step, we solve Eqs. (4.22) and (4.23) for ~y and ~m, respectively, which allowsexpressing the dependent quantities as linear combinations of the independent quantities(~xR and ~nR) through the so-called link matrix L [7] de�ned by

~y = ~s0 + L · ~xR (4.27)

~m = n0 + L · ~nR . (4.28)

Inserting the decompositions from Eq. (4.26) into Eq. (4.28) and comparing with Eq.(4.27) shows that dependent and independent �uctuations are related as

~η = L · ~ξR. (4.29)

Finally, one obtains the dynamics of the reduced system in the form

d~xRdt

= NR · ~f0 (~xR;~s0) (4.30)

26


where the reduced stoichiometric matrix NR is obtained by taking the �rst k′ rows of thematrix P ·N and the reaction rate vector ~f0 is evaluated at the conservation relationsusing Eqs. (4.24) and (4.27). Similarly, we de�ne the reduced drift matrix KR and thereduced di�usion matrix DR by

KR,ii′ (~xR;~s0) :=r∑j=1

NR,ij∂f0,j (~xR;~s0)

∂xR,i′and (4.31)

DR,ii′ (~xR;~s0) :=r∑j=1

f0,j (~xR;~s0)NR,ijNR,i′j

where the indices range between i, i′ = 1, . . . , k′. Using KR and DR the dynamics of thevariance-covariance matrix of the reduced system is determined by

dσRdt

= KR · σR + σR ·KTR + DR (4.32)

where the components of σR are given by σR,ii′ =⟨ξR,iξR,i′

⟩for i, i′ = 1, . . . , k′. Hence,

one can apply the linear noise approximation to the reduced system, de�ned by Eq.(4.30), in a similar way as described in Section 4.2.The components of the variance-covariance matrix associated with the conserved species

can be calculated from the reduced variance-covariance matrix σR and Eq. (4.29). Specif-ically, one obtains

σ′(k′+α)(k′+β) = 〈ηαηβ〉 =

⟨k′∑i=1

LαiξR,i

k′∑i′=1

Lβi′ξR,i′

⟩

=

k′∑i,i′=1

LαiLβi′σR,ii′ , α, β = 1, . . . , k − k′ (4.33)

where σ′ denotes the variance-covariance matrix with respect to the transformed �uc-tuation vector ~ξ′ de�ned in Eq. (4.25). Similarly, the `mixed' components are obtainedfrom

σ(k′+α)(i) = 〈ηαξR,i〉 =

⟨(k′∑i′=1

Lαi′ξR,i′

)ξR,i

⟩

=

k′∑i′=1

Lαi′σR,i′i, i = 1, . . . , k′, α = 1, . . . , k − k′ . (4.34)

4.5. Examples

In the next two subsections we demonstrate how to apply the linear noise approximationto speci�c reaction systems. In both cases it is possible to derive explicit expressions forthe steady state �uctuations. In addition, the example in subsection 4.5.2 shows how todeal with mass-conservation relations.

27


4.5.1. Noise Reduction Through Receptor Dimerization

As a �rst example, we consider the reaction scheme

∅ks�kd1

R (X1) (4.35)

R (X1) +R (X1)k+

�k−

R2 (X2) (4.36)

R2 (X2)kd2→ ∅ (4.37)

which describes synthesis and degradation of a receptor R (Eq. 4.35) and its dimerization(Eq. 4.36). In addition, we assume that the dimerized receptor (R2) can also be degradedwith a speci�c rate that may di�er from that for the monomer (4.37). First, we notethat in the absence of dimerization independent synthesis and degradation of receptormolecules, as described by Eq. (4.35), leads to Poissonian steady state �uctuations wherethe average concentration 〈R〉 = ks/kd1 is equal to its variance σ

2R, so that the Fano factor

F1 = σ2R/ 〈R〉 becomes 1. Moreover, Hayot and Jayaprakash observed [5] that, withinthe LNA, this holds also true for receptor dimers, i.e. F2 = σ2R2

/ 〈R2〉 = 1. Here, weshow that a non-vanishing dimer degradation rate (kd2 6= 0) leads to sub-Poissonian�uctuations for both, monomers and dimers. In addition, we �nd that the steady state�uctuations of dimerized receptors can become lower than those for monomers (F2 < F1)under certain conditions.The stoichiometric matrix for the system in Eqs. (4.35) - (4.37) is given by

N =RR2

(1 −1 −2 2 00 0 1 −1 −1

). (4.38)

Hence, there are �ve stoichiometric vectors

~N1 =

(10

), ~N2 =

(−10

), ~N3 =

(−21

), ~N4 =

(2−1

), ~N5 =

(0−1

),

but no conservation relations.Let's denote the number of receptor monomers and dimers by n1 and n2, respectively.

Then the �ve transition rates are given by

W1

(~n|~n− ~N1

)= k̄s, W2

(~n|~n− ~N2

)= kd1n1

W3

(~n|~n− ~N3

)= k̄+n1 (n1 − 1) , W4

(~n|~n− ~N4

)= k−n2

W5

(~n|~n− ~N5

)= kd2n2

where ~n = (n1, n2)T . The units of k̄s and k̄

+ are

[k̄s] =particles

sand [k̄+] =

1

particles · s

28


whereas the unit of the remaining three constants is

[kd1] = [k−] = [kd2] =1

s.

Next, we expand the transition rates in the form (cf. Eq. 4.7)

Wj (~n) = Vmf0,j (~x) + f1,j (~x) + . . . , ~x =~n

Vm.

For W2, W4 and W5 this yields

W2 = Vmkd1x1, W4 = Vmk−x2 and W5 = Vmkd2x2

which shows that

f0,2 (~x) = kd1x1, f0,4 (~x) = k−x2 and f0,5 (~x) = kd2x2 (4.39)

and fl,j (~x) ≡ 0 for l ≥ 1 and j = 2, 4, 5. Hence, �rst-order rate constants are una�ectedwhen going from a mesoscopic to a macroscopic description.However, expansion of the remaining transition rates

W1 = Vmks and W3 = k̄+Vmx1 (Vmx1 − 1) ≡ Vmk+x21 − k+x1 (4.40)

shows that zero-order rate constants become inversely proportional to the reaction volume(ks = k̄s/Vm) whereas second-order rate constants become proportional to the reactionvolume (k+ = Vmk̄

+) so that ks and k+, as de�ned in Eqs. (4.35) and (4.36), now have

typical `macroscopic' units

[ks] =mol

liter · s≡ M

sand

[k+]

=liter

mol · s≡ 1

M · s.

From Eq. (4.40) we infer that

f0,1 (~x) = ks and f0,3 (~x) = k+x21 (4.41)

with fl,1 (~x) ≡ 0 for l ≥ 1 and fl,3 (~x) ≡ 0 for l ≥ 2. We also see that, as a result of thedimerization reaction, there appears an O (1)-term (f1,3 (~x) = −k̄+x1) in the expansionfor W3 which is, however, unimportant for the linear noise approximation.

The functions f(0)j , de�ned in Eqs. (4.39) and (4.41), are used to construct the reaction

rate vector~f0 =

(ks, kd1x1, k

+x21, k−x2, kd2x2

)T. (4.42)

Then, the dynamics of the average concentration ~x = (x1, x2)T is determined by the

ODE system (cf. Eq. 4.8)d~x

dt= N · ~f0 (~x)

29


or, in components,

dx1dt

= ks − kd1x1 − 2k+x21 + 2k−x2 (4.43)

dx2dt

= k+x21 −(k− + kd2

)x2 .

For the matrices K and D, de�ned in Eq. (4.10), we obtain

K =

(−kd1 − 4k+x1 2k−

2k+x1 − (k− + kd2)

)(4.44)

and

D = N · diag(~f0

)·NT

=

(ks + kd1x1 + 4k−x2 + 4k+x21 −2k−x2 − 2k+x21

−2k−x2 − 2k+x21 k+x21 + k−x2 + kd2x2

)(4.45)

where diag(~f0

)is a diagonal matrix which has the components f0j on its diagonal.

The solution of the steady state equations

ks − kd1x1 − 2k+x21 + 2k−x2 = 0 (4.46)

k+x21 −(k− + kd2

)x2 = 0 ,

which result from Eq. (4.43) by setting dx1/dt = 0 = dx2/dt, is given by

xs2 =1

2α (Rs − xs1) and (4.47)

xs1 =γ

2

(√1 +

4Rsγ− 1

)(4.48)

where

Rs =kskd1

, α =kd1kd2

, β =kd2

k−, KD =

k−

k+(4.49)

andγ ≡ KD (1 + β)

α

2.

Note that in the limit 4Rs � γ expansion of the square root in Eq. (4.48) yields toleading order

xs1 ≈ Rs and xs2 ≈R2s

KD (1 + β). (4.50)

Hence, by increasing the dissociation constant KD (weak association of monomers) orincreasing β (fast dimer degradation) one recovers the case of simple receptor synthesisand degradation described by Eq. (4.35) where xs1 ≈ Rs and xs2 ≈ 0. However, in generalone expects that dimer dissociation is much faster than dimer degradation so that β � 1.

30


To �nd the three independent components of the variance-covariance matrix

σ =

(σ11 σ12σ12 σ22

)we have to solve the system of linear equations de�ned by

K (~xs) · σ + σ ·K (~xs)T + D (~xs) = 0 (4.51)

where the matrices K and D, de�ned in Eqs. (4.44) and (4.45), are now evaluated atthe steady state (cf. Eqs. 4.47 and 4.48). The solution of Eq. (4.51) is straightforwardalthough it is typically not easy to interpret. However, the resulting expressions can besubstantially simpli�ed if one applies algebraic manipulations while repeatedly makinguse of the steady state relations in Eq. (4.46). The result is

σs11 = xs1

(1− (1 + β (1 + α))xs1/KD + 4βxs2/KD

(1 + β (1 + α) + 4xs1/KD) (α (1 + β) + 4xs1/KD)

)(4.52)

σs22 = xs2

(1− (2xs1/KD)2

(1 + β (1 + α) + 4xs1/KD) (α (1 + β) + 4xs1/KD)

)(4.53)

σs12 = −2(1 + β)xs2x

s1/KD

(1 + β (1 + α) + 4xs1/KD) (α (1 + β) + 4xs1/KD). (4.54)

From Eqs. (4.52) and (4.53) we see that receptor dimerization leads to sub-Poissonian�uctuations for both receptor monomers (F1 = σs11/x

s1 < 1) and receptor dimers (F2 =

σs22/xs2 < 1). Similarly, Eq. (4.54) tells us that �uctuations in receptor monomers are

always anti-correlated with �uctuations in receptor dimers since σs12 < 0. This is, ofcourse, not surprising since an increase of the dimer concentration must be accompaniedby a decrease in the concentration of monomers. However, since a decrease in dimersdoes not necessarily result from an increase in monomers, but could also result from adegradation of dimers, the strength of the anti-correlation will vary with parameters.To estimate how much the intrinsic noise can be reduced by receptor dimerization we

consider the special case where α = 1 (monomers and dimers are degraded at the samerate) and β � 1. In that case the Fano factors can be approximated by

F1 ≈ 1− 1

4

4xs1/KD

(1 + 4xs1/KD)2(4.55)

F2 ≈ 1− 1

4

(4xs1/KD)2

(1 + 4xs1/KD)2(4.56)

which implies that

F2 < F1, for xs1 >KD

4.

In particular, one can show that, in the limit xs1 � KD/4, F1 → 1 whereas F2 → 3/4.Hence, dimerization can reduce the steady state �uctuations of dimerized receptors if Rsis su�ciently large. Indeed, xs1 ∼

√γRs for Rs � γ/4 (cf. Eq. 4.48).

31


Finally, we note that the case kd2 = 0, studied in Ref. [5], is recovered by lettingin Eqs. (4.52) - (4.54) α → ∞ and β → 0 while keeping the product α · β = kd1/k

−

constant (cf. Eq. 4.49) which leads to xs1 → Rs and xs2 → R2s/KD (cf. Eq. 4.50) so

that �uctuations in monomers and dimers become Poissonian (σs11 → xs1, σs22 → xs2) and

correlations between monomers and dimers vanish (σs12 → 0).

4.5.2. Receptor-Ligand Binding with Mass Conservation

As a second example, we consider the simple reaction scheme

L (X1) +R (X2)k+1�k−1

R-L (X3) (4.57)

which describes the binding of a ligand (L) to a receptor (R) under in vitro conditions.The transition rates are given by

W1 = k̄+1 n1n2 and W2 = k−1 n3 (4.58)

where k̄+1 and k−1 have the units 1/ (particle · s) and 1/s, respectively. The stoichiometric

matrix of the system in Eq. (4.57) reads

N =RLR-L

−1 1−1 11 −1

.

In this example, we neglect reactions for synthesis and degradation of the ligand andreceptor molecules so that their total numbers remain constant. Consequently, two ofthe three rows of N are linearly dependent. A positive basis of the left null space of Nis given by

S =

(1 0 10 1 1

)and the corresponding conservation relations read

S · ~x (t) =

(x1 (t) + x3 (t)x2 (t) + x3 (t)

)= ~s0 ≡

(LTRT

). (4.59)

We choose X3 as the independent, and X1 and X2 as the dependent species, i.e. we set(cf. Eqs. 4.26) xR

y1y2

= P ·

x1x2x3

=

x3x1x2

(4.60)

where P denotes the permutation matrix (cf. Eq. 4.24)

P =

0 0 11 0 00 1 0

.

32


The �uctuations are rearranged as ξRη1η2

= P ·

ξ1ξ2ξ3

=

ξ3ξ1ξ2

. (4.61)

Using Eq. (4.59) we can express the dependent species concentrations as a function ofthe independent species concentration through

y1 = LT − xR (4.62)

y2 = RT − xR

or~y = ~s0 + L · ~xR

where the (2× 1) link matrix L, as de�ned in Eq. (4.27), is given by

L =

(−1−1

). (4.63)

Hence, �uctuations of the dependent species can be obtained from that of the independentspecies through (cf. Eq. 4.29)

η1 = −ξR (4.64)

η2 = −ξR .

The dynamics of the reduced system is then determined by

dxRdt

= NR · ~f0 (xR, ~s0) = k+1 (LT − xR) (RT − xR)− k−1 xR (4.65)

where the reduced stoichiometric matrix is obtained from the �rst row of P ·N as

NR =(

1 −1).

The reaction rate vector is obtained by �rst expanding the transition rates, de�ned inEq. (4.58), as

W1 = Vmk+1 x1x2 ≡ Vmk

+1 y1y2

W2 = Vmk−1 x3 ≡ Vmk

−1 xR

and then using the conservation relations in Eqs. (4.62) to replace y1 and y2 by xR and~s0 = (LT , RT )T leading to

~f0 (xR, ~s0) =

(k+1 (LT − xR) (RT − xR)

k−1 xR

).

In Eq. (4.65) the second-order rate constant k+1 has the unit liter/ (mol · s) ≡ 1/ (M · s).

33


Since the reduced system is 1-dimensional there is only 1 independent component ofthe variance-covariance matrix which is determined by

σR = − DR

2KR

where σR = 〈ξRξR〉 = 〈ξ3ξ3〉 ≡ σ33 (cf. Eq. 4.61). The reduced Jacobian and di�usionmatrices (KR and DR) are given by

KR = −k+1 (LT +RT − 2xR)− k−1DR = k+1 (LT − xR) (RT − xR) + k−1 xR.

Hence, under steady state conditions we have

k+1 (LT − xR) (RT − xR)!

= k−1 xR

and the variance of the receptor-ligand complex can be written as

σR =1

2

k+1 (LT − xR) (RT − xR) + k−1 xR

k+1 (LT +RT − 2xR) + k−1

=xR

LT +RT − 2xR +KD(4.66)

where xR is determined by the quadratic equation

x2R − xR (LT +RT +KD) + LTRT = 0 (4.67)

and KD = k−1 /k+1 denotes the dissociation constant of the receptor-ligand complex. The

biologically feasable solution of Eq. (4.67) is given by

xR =LT +RT +KD

2

(1−

√1− 4

LTRT

(LT +RT +KD)2

). (4.68)

The remaining components of the variance-covariance matrix can be calculated usingthe expressions in Eqs. (4.61) and (4.64) which follow from the general expressions inEqs. (4.33) and (4.34). For example, the variances of ligand and free receptors are

σ11 = 〈ξ1ξ1〉 = 〈η1η1〉 = 〈(−ξR) (−ξR)〉 = σR

σ22 = 〈ξ2ξ2〉 = 〈η2η2〉 = 〈(−ξR) (−ξR)〉 = σR.

Hence, the variances of all three species are the same (σ11 = σ22 = σ33 ≡ σR). Similary,the calculation of the covariances yields

σ12 = 〈ξ1ξ2〉 = 〈η1η2〉 = 〈(−ξR) (−ξR)〉 = σR

σ13 = 〈ξ1ξ3〉 = 〈η1ξR〉 = 〈(−ξR) ξR〉 = −σRσ23 = 〈ξ2ξ3〉 = 〈η2ξR〉 = 〈(−ξR) ξR〉 = −σR.

34


This shows that the correlation coe�cients, as de�ned in Eq. (4.18), are given by

r13 = r23 = −1 and r12 = 1

which expresses the obvious fact that the disappearance of a ligand or receptor moleculeis always accompanied by the appearance of a receptor-ligand complex.To obtain explicit expressions for other stochastic quantities such as the Fano factor

one may use the expressions for xR and σR in Eqs. (4.68) and (4.66). Alternatively,one can try to obtain simpler expressions for xR from suitable approximate solutions ofEq. (4.67). Speci�cally, if KD � RT or KD � LT the solution of Eq. (4.67) can beapproximated by the discontinuous function

xR ≈

LT

(1− KD

RT−LT

)RT

(1− KD

LT−RT

) LT < RT

RT < LT .(4.69)

In the opposite case, when KD � RT or KD � LT one obtains

xR ≈LTRT

LT +RT +KD. (4.70)

Hence, in the �rst case (Eq. 4.69) the Fano factors are given to leading order by

F1 =σR

LT − xR≈

{1

RTKD

(LT−RT )2

LT < RT

RT < LT(4.71)

F2 =σR

RT − xR≈

{LTKD

(RT−LT )2

1

LT < RT

RT < LT

F3 =σRxR≈

{KD

RT−LTKD

LT−RT

LT < RT

RT < LT ,

whereas in the second case (Eq. 4.70) they are given to leading order by

F1 ≈ KDRT(LT +KD) (LT +RT +KD)

(4.72)

F2 ≈ KDLT(RT +KD) (LT +RT +KD)

F3 ≈ KD

LT +RT +KD.

From the expressions in Eqs. (4.71) we see that if receptor-ligand binding is tight, sothat KD � min (RT , LT ), F1 exhibits a switch-like transition from F1 ≈ 1 for LT < RT(Poissonian noise) to F1 ≈ 0 for LT > RT (noise suppression) and vice versa for F2.

Problem

Check the validity of the approximate expressions in Eqs. (4.69) - (4.72) with LiNA!

35

A. Derivation of the Linear Noise

Approximation

In the following, we give a derivation of the dynamic equations for the mean-�eld ~x (Eq.

4.8) and the probability density of the �uctuations Π(~ξ, t)(Eq. 4.9), which are derived

from the master equation (Eq. 4.3) using the decomposion of the particle number vectorin Eq. (4.4). The derivation is a multivariate extension of that given in Ref. [9] for thespecial case of biochemical reaction networks where the jump sizes in the transition ratesare determined by the entries of the stoichiometric matrix (cf. Eq. 4.3). An alternativederivation of the multivariate linear noise approximation, based on the step operatorformalism, can be found in Ref. [1]. A derivation that goes beyond the linear noiseapproximation has been given by Grima in Ref. [4].

A.1. Transformation of the Master Equation

The probability densities P (~n, t) and Π(~ξ, t)are related through

P (~n, t) = P

(Vm~x+ V

12m~ξ, t

)= V

− 12

m Π(~ξ, t)

(A.1)

which follows from the invariance of the probability measure

P (~n, t) d~n = P (~n, t)V12md~ξ

!= Π

(~ξ, t)d~ξ

and Eq. (4.4). The probability density at the shifted argument can be obtained from

P(~n− ~Nj , t

)= P

(Vm~x+ V

12m

(~ξ − V −

12

m~Nj

), t

)= V

− 12

m Π

(~ξ − V −

12

m~Nj , t

). (A.2)

36

A. Derivation of the Linear Noise Approximation

Under the transformation in Eq. (4.4) the time-derivative in Eq. (4.3) becomes

dP (~n, t)

dt= V

− 12

m

∂Π(~ξ, t)

∂t

~n

= V− 1

2m

∂Π(~ξ, t)

∂t

~ξ

+k∑i=1

∂Π(~ξ, t)

∂ξi

t

(dξidt

)~n

= V

− 12

m

∂Π(~ξ, t)

∂t

~ξ

− V12m

k∑i=1

∂Π(~ξ, t)

∂ξi

t

dxidt

(A.3)

where the last step follows from Eq. (4.4) if ~n is kept constant.Next, we have to transform the transition rates in Eq. (4.3). To this end we recall the

expansion from Eq. (4.7)

Wj (~n) =∞∑l=0

V 1−lm fl,j

(~x+ V

− 12

m~ξ

)which can be used to obtain the transition rates at the shifted argument as

Wj

(~n− ~Nj

)=∞∑l=0

V 1−lm fl,j

(~x+ V

− 12

m

(~ξ − V −

12

m~Nj

)). (A.4)

Inserting the expressions in Eqs. (A.1) - (A.4) into the master equation (Eq. 4.3) yields

(after multiplication by V12m )

∂Π(~ξ, t)

∂t− V

12m

k∑i=1

∂Π(~ξ, t)

∂ξi

dxidt

= (A.5)

∞∑l=0

V 1−lm

r∑j=1

[fl,j

(~x+ V

− 12

m

(~ξ − V −

12

m~Nj

))Π

(~ξ − V −

12

m~Nj , t

)(A.6)

− fl,j(~x+ V

− 12

m~ξ

)Π(~ξ, t)]. (A.7)

The idea of van Kampen's system size expansion is to expand the right-hand side of this

master equation in powers of V− 1

2m . Note that the left-hand side of the master equation

(Eq. A.5) has already the form of an expansion in powers of V− 1

2m ≡ ε which consists of

terms of O(ε0)and O

(ε−1). In order to match these terms by corresponding terms on

the right-hand side of the master equation (Eqs. A.6 and A.7) we expand the functions

fl,j

(~x+ V

− 12

m

(~ξ − V −

12

m~Nj

))Π

(~ξ − V −

12

m~Nj , t

)

37


with respect to V− 1

2m

~Nj in the argument ~ξ − V −12

m~Nj which leads to

fl,j

(~x+ V

− 12

m

(~ξ − V −

12

m~Nj

))Π

(~ξ − V −

12

m~Nj , t

)= fl,j

(~x+ V

− 12

m~ξ

)Π(~ξ, t)

(A.8)

− V− 1

2m

k∑i=1

∂

[fl,j

(~x+ V

− 12

m~ξ

)Π(~ξ, t)]

∂ξiNij (A.9)

+1

2V −1m

k∑i,i′=1

∂2[fl,j

(~x+ V

− 12

m~ξ

)Π(~ξ, t)]

∂ξi∂ξi′NijNi′j +O

(V− 3

2m

)(A.10)

Note that the lowest order term in Eq. (A.8) cancels with the term in Eq. (A.7) of

the master equation. The expressions in Eqs. (A.9) and (A.10) are of O(V− 1

2m

)and

O(V −1m

), respectively, so that they contribute terms of O

(V

12−l

m

)and O

(V −lm

)to the

master equation. However, since in Eq. (A.5) the lowest order term is of O (1) the termscoming from Eqs. (A.9) and (A.10) can only be matched if l = 0.

The arguments of the functions fl,j in Eqs. (A.9) and (A.10) still depend on V− 1

2m

which, after expansion with respect to V− 1

2m

~ξ

fl,j

(~x+ V

− 12

m~ξ

)= fl,j (~x) + V

− 12

m

k∑i=1

∂fl,j (~x)

∂xiξi +O

(V −1m

), (A.11)

can potentially contribute to the matching with Eq. (A.5). However, for the second order

term in Eq. (A.10) the higher order terms in Eq. (A.11) would yield terms of O(V− 3

2m

)and higher which, thus, cannot be matched with the terms in Eq. (A.5). Hence, we maywrite Eq. (A.10) in the form

1

2V −1m

k∑i,i′=1

∂2[fl,j

(~x+ V

− 12

m~ξ

)Π(~ξ, t)]

∂ξi∂ξi′NijNi′j

=1

2V −1m

k∑i,i′=1

fl,j (~x)NijNi′j

∂2Π(~ξ, t)

∂ξi∂ξi′+O

(V− 3

2m

). (A.12)

In contrast, for the �rst order term in Eq. (A.9), the �rst two terms in Eq. (A.11) give a

38


contribution so that we write this term in the form

V− 1

2m

k∑i=1

∂

[fl,j

(~x+ V

− 12

m~ξ

)Π(~ξ, t)]

∂ξiNij

= V− 1

2m

k∑i=1

∂Π(~ξ, t)

∂ξiNijfl,j (~x)

+ V −1m

k∑i,i′=1

∂fl,j (~x)

∂xi′

∂[ξi′Π

(~ξ, t)]

∂ξiNij +O

(V− 3

2m

). (A.13)

Finally, using the expressions in Eqs. (A.12) and (A.13) in Eqs. (A.10) and (A.9) and,the resulting expression in Eq. (A.6) gives

∂Π(~ξ, t)

∂t− V

12m

k∑i=1

∂Π(~ξ, t)

∂ξi

dxidt

= −V12m

r∑j=1

k∑i=1

∂Π(~ξ, t)

∂ξiNijf0,j (~x) (A.14)

−r∑j=1

k∑i,i′=1

Nij∂f0,j (~x)

∂xi′

∂[ξi′Π

(~ξ, t)]

∂ξi

+1

2

r∑j=1

f0,j (~x)

k∑i,i′=1

NijNi′j

∂2Π(~ξ, t)

∂ξi∂ξi′+O

(V− 1

2m

)

where we have retained only the l = 0 term.

A.2. E�ective Equations for ~x (t) and Π(~ξ, t)

Comparing �rst the terms of highest order in Eq. (A.14) gives

O(V

12m

):

k∑i=1

∂Π(~ξ, t)

∂ξi

dxidt−

r∑j=1

Nijf0,j (~x)

= 0 .

This implies that ~x has to ful�ll the mean-�eld equations (cf. Eq. 4.8)

dxidt

=

r∑j=1

Nijf0,j (~x) , i = 1, . . . , k (A.15)

provided that

∂Π(~ξ, t)

∂ξi6= 0, i = 1, . . . , k

39


which means that Π must not be constant with respect to ~ξ or, in other words, themean-�eld equation (Eq. A.15) is only valid if there are �uctuations in the system.Comparing the terms at next order yields a Fokker-Planck equation for the density of

the �uctuations given by

O (1) :∂Π(~ξ, t)

∂t= −

r∑j=1

k∑i,i′=1

Nij∂f0,j (~x)

∂xi′

∂[ξi′Π

(~ξ, t)]

∂ξi(A.16)

+1

2

r∑j=1

f0,j (~x)

k∑i,i′=1

NijNi′j

∂2Π(~ξ, t)

∂ξi∂ξi′.

Introducing the drift matrix K and the di�usion matrix D by

Kii′ (~x) :=r∑j=1

Nij∂f0,j (~x)

∂xi′and Dii′ (~x) :=

r∑j=1

f0,j (~x)NijNi′j (A.17)

the Fokker-Planck equation can be written in compact notation

∂Π(~ξ, t)

∂t=

k∑i,i′=1

−Kii′ (~x)∂[ξi′Π

(~ξ, t)]

∂ξi+

1

2Dii′ (~x)

∂2Π(~ξ, t)

∂ξi∂ξi′

(A.18)

which agrees with Eq. (4.9).

A.3. Equations for 〈ξi (t)〉 and σii′ (t) = 〈ξiξi′〉With the help of the Fokker-Planck equation (A.18) one can derive equations for the

temporal evolution of the average of a function f(~ξ)according to the scheme

d⟨f(~ξ)⟩

dt=

d

dt

∫ ∞−∞

dξ1 · . . . ·∫ ∞−∞

dξk f(~ξ)

Π(~ξ, t)

(A.19)

= Πki=1

∫ ∞−∞

dξi f(~ξ) ∂Π

(~ξ, t)

∂t

=k∑

j,j′=1

∫d~ξ f

(~ξ)−Kjj′

∂[ξj′Π

(~ξ, t)]

∂ξj+

1

2Djj′

∂2Π(~ξ, t)

∂ξj∂ξj′

(A.20)where we have introduced the notation∫

d~ξ ≡∫ ∞−∞

dξ1 · . . . ·∫ ∞−∞

dξk = Πki=1

∫ ∞−∞

dξi .

40


Also, in order to rewrite the expressions in Eq. (A.20) in terms of average quantities wewill have to do partial integrations of the form∫

d~ξ f(~ξ) ∂

∂ξi

[g(~ξ)

Π(~ξ, t)]

= −∫d~ξ g

(~ξ)

Π(~ξ, t) ∂

∂ξif(~ξ)

where the boundary term vanishes since Π(~ξ, t), being a multivariate Gaussian (Eq.

4.11), decays exponentially as∣∣∣~ξ∣∣∣→∞.

For the mean value we obtain

d 〈ξi〉dt

=k∑

j,j′=1

∫d~ξ ξi

−Kjj′

∂[ξj′Π

(~ξ, t)]

∂ξj+

1

2Djj′

∂2Π(~ξ, t)

∂ξj∂ξj′

=

k∑j,j′=1

∫d~ξ

(Kjj′ξj′Π

(~ξ, t) ∂ξi∂ξj

+1

2Djj′Π

(~ξ, t) ∂2ξi∂ξj∂ξj′

)

=k∑

j,j′=1

∫d~ξ Kjj′ξj′Π

(~ξ, t)δij

=k∑

,j′=1

Kij′⟨ξj′⟩

or, in vector notation,

d⟨~ξ⟩

dt= K ·

⟨~ξ⟩, (A.21)

i.e. the �uctuations ~ξ obey the same equation as small deviations from a steady state ofthe mean-�eld equations (Eq. A.15). Hence, if K (~x) is evaluated at an asymptoticallystable steady state all eigenvalues of K have a negative real part so that the �uctuationsdecay to zero in time.

41


The calculation for the variance-covariance matrix σii′ = 〈ξiξi′〉 yields

d 〈ξiξi′〉dt

=

k∑j,j′=1

∫d~ξ ξiξi′

−Kjj′

∂[ξj′Π

(~ξ, t)]

∂ξj+

1

2Djj′

∂2Π(~ξ, t)

∂ξj∂ξj′

=

k∑j,j′=1

∫d~ξ

(Kjj′ξj′Π

(~ξ, t) ∂ (ξiξi′)

∂ξj+

1

2Djj′Π

(~ξ, t) ∂2 (ξiξi′)

∂ξj∂ξj′

)

=k∑

j,j′=1

∫d~ξ Kjj′ξj′Π

(~ξ, t) (δijξi′ + δi′jξi

)+

1

2

k∑j,j′=1

∫d~ξ Djj′Π

(~ξ, t) (δijδi′j′ + δi′jδij′

)=

k∑j′=1

Kij′⟨ξj′ξi′

⟩+

k∑j′=1

Ki′j′⟨ξj′ξi

⟩+

1

2(Dii′ +Di′i)

or, in matrix notation,dσ

dt= K · σ + σ ·KT + D (A.22)

where we have used the fact that D is symmetric (Dii′ = Di′i).

42

Bibliography

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[3] A. Goldbeter and D. E. Koshland Jr., An ampli�ed sensitivity arising fromcovalent modi�cation in biological systems, Proc. Natl. Acad. Sci. USA, 78 (1981),pp. 6840�6844.

[4] R. Grima, An e�ective rate equation approach to reaction kinetics in small volumes:Theory and application to biochemical reactions in nonequilibrium steady-state con-ditions, J. Chem. Phys., 133 (2012), p. 035101.

[5] F. Hayot and C. Jayaprakash, The linear noise approximation for molecular�uctuations within cells, Phys. Biol., 1 (2004), pp. 205�210.

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[7] C. Reder, Metabolic control theory: A structural approach, J. Theor. Biol., 135(1988), pp. 175�201.

[8] R. Straube and C. Conradi, Reciprocal enzyme regulation as a source of bistabilityin covalent modi�cation systems, J. Theor. Biol., 313 (2013), pp. 56�74.

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LiNA A Graphical Matlab Tool for Analyzing Intrinsic …LiNA A Graphical Matlab Tool for Analyzing Intrinsic Noise in Biochemical Reaction Networks Tutorials and Theoretical Background

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