An extended model for the repression of photosynthesis genes by the AppA/PpsR system in Rhodobacter sphaeroides

An extended model for the repression of photosynthesisgenes by the AppA/PpsR system inRhodobacter sphaeroidesRakesh Pandey1, Dietrich Flockerzi2, Marcus J. B. Hauser3 and Ronny Straube1

1 Analysis and Redesign of Biological Networks Group, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg,

Germany

2 Systems and Control Theory Group, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

3 Biophysics Group, Institute of Experimental Physics, Otto-von-Guericke University, Magdeburg, Germany

Keywords

bistability; photosynthetic bacteria; purple

nonsulfur bacteria; signal transduction;

steady-state analysis

Correspondence

R. Straube, Analysis and Redesign of

Biological Networks Group, Max Planck

Institute for Dynamics of Complex Technical

Systems, Sandtorstrasse 1, Magdeburg

D-39106, Germany

Fax: +49 391 6110543

Tel: +49 391 6110481

E-mail: [email protected]

(Received 30 November 2011, revised 31

January 2012, accepted 3 Febraury 2012)

doi:10.1111/j.1742-4658.2012.08520.x

Purple bacteria derive energy from aerobic respiration or photosynthesis

depending on the availability of oxygen and light. Under aerobic condi-

tions, photosynthesis genes are specifically repressed by the PpsR protein.

In Rhodobacter sphaeroides, the repressive action of PpsR is antagonized

by the blue-light and redox-sensitive flavoprotein AppA, which sequesters

PpsR under anaerobic conditions into transcriptionally inactive complexes.

However, under semi-aerobic conditions, blue-light excitation of AppA

causes the AppA–PpsR complexes to dissociate, again leading to a repres-

sion of photosynthesis genes. We have recently developed a simple mathe-

matical model suggesting that this phenotype arises from the formation of

a maximum in the response curve of reduced PpsR at intermediate oxygen

concentrations. However, this model focused mainly on the oxygen-depen-

dent interactions whereas light regulation was only implemented in a sim-

plified manner. In the present study, we incorporate a more detailed

mechanism for the light-dependent interaction between AppA and PpsR,

which now allows for a direct comparison with experiments. Specifically,

we take into account that, upon blue–light excitation, AppA undergoes a

conformational change, creating a long-lived signalling state causing the

dissociation of the AppA–PpsR complexes. The predictions of the extended

model are found to be in good agreement with experimental results on the

light-dependent repression of photosynthesis genes under semi-aerobic con-

ditions. We also identify the potential kinetic and stoichiometric constraints

that the interplay between light and redox regulation imposes on the func-

tionality of the AppA/PpsR system, especially with respect to a possible

bistable response.

Introduction

Purple non-sulfur bacteria can generate energy in the

form of ATP through both respiration and photosyn-

thesis (PS) depending on the availability of oxygen and

light. In Rhodobacter sphaeroides, the transition from

aerobic to anaerobic photosynthetic growth is gov-

erned by three regulatory systems acting at the tran-

scriptional level: the PrrB/PrrA two component system

[1,2], the anaerobic activator Fnrl [3,4] and the aerobic

repressor PpsR [5,6]. Compared to the first two sys-

tems, which are global gene regulatory systems, PpsR

is specifically involved in the regulation of PS genes

such as bacteriochlorophyll synthesis (bch), carotenoid

Abbreviations

BLUF, blue-light sensing using FAD; ODE, ordinary differential equation; PS, photosynthesis.

FEBS Journal (2012) ª 2012 The Authors Journal compilation ª 2012 FEBS 1

synthesis (crt), pigment-binding proteins of the light

harvesting complex II (puc) and polypeptides of the

reaction centers (puf ) [7,8].

PpsR exists as a stable tetramer in solution [9].

Under aerobic conditions ([O2] � 200 lm dissolved

oxygen concentration), it binds cooperatively to a pal-

indromic sequence in the target promoters of PS genes

[9–11]. DNA-binding of PpsR is stimulated by oxygen

through the formation of an intramolecular disulfide

bond between two redox-active cysteine residues [6,12].

Under anaerobic conditions or for a low concentration

of dissolved oxygen ([O2] £ 3 lm), the disulfide bond is

reduced to a thiol group, which results in a lower

DNA-binding affinity of the reduced form of PpsR

compared to its oxidized state [9]. Experiments have

also shown that the reduction of PpsR is mediated by

the oxygen- and blue-light-sensitive flavoprotein

AppA, which appears to be unique to R. sphaeroides.

AppA utilizes the two cofactors FAD and heme to

sense blue-light and oxygen, respectively. Whereas

FAD is noncovalently attached to the N-terminal blue-

light sensing using FAD (BLUF) domain of AppA

[13–15], the heme cofactor associates to a region in the

C-terminal part of that protein [16,17].

Besides being involved in the reduction of PpsR, the

AppA protein also forms transcriptionally inactive

complexes with the reduced form of PpsR under

anaerobic conditions [9]. In this process, AppA is con-

sidered to sequester PpsR from the DNA, which con-

tributes to the induction of PS genes in the absence of

oxygen [18]. Interestingly, in the presence of blue-light

illumination (450 nm), AppA undergoes a conforma-

tional change [19,20], which presumably leads to the

dissociation of the AppA–PpsR complex [16]. As a

result of this oxygen- and light-dependent interaction,

R. sphaeroides exhibits a unique phenotype under

semi-aerobic conditions ([O2] � 100 lm) where PS

genes are repressed by sufficiently intense blue-light

irradiance (LI ‡ 0.2 lmolÆm)2Æs)1) [21–23].We have recently developed a simple mathematical

model [24] for the oxygen- and light-dependent interac-

tion between AppA and PpsR based on the two mech-

anisms proposed by Masuda and Bauer [9]: (a) the

AppA-mediated reduction of a disulfide bond in PpsR

and (b) the light-inhibited formation of an AppA–

PpsR complex. A steady-state analysis of the model

equations showed that a non-monotonic dependence

of the concentration of reduced PpsR on the oxygen

concentration with a light-dependent maximum at

intermediate oxygen levels can arise when the rate of

PpsR reduction is much larger than that of AppA

reduction. We argued that this could provide a qualita-

tive explanation for the observed phenotype of high

light repression of PS genes at intermediate oxygen lev-

els if the oxygen concentration, where the maximum

occurs, is identified with the semi-aerobic regime. In

addition, we found that the light-dependent complex

formation between AppA and PpsR can provide an

implicit positive feedback loop, which may result in a

bistable response as a result of changing light and oxy-

gen conditions.

In the previously developed model [24], we have

incorporated the light-dependent regulation of the

interaction between AppA and PpsR only in an effec-

tive manner. Specifically, we have assumed that the

association between AppA and PpsR is directly inhib-

ited by light. However, experiments suggest a more

elaborated mechanism [19,20,16] involving a light-

induced conformational change of the AppA protein,

which may affect the interaction between AppA and

PpsR. In particular, it is conceivable that this confor-

mational change may induce the dissociation of the

AppA–PpsR complex that is predominantly formed

under anaerobic conditions in the absence of light. In

the present study, we investigate the conditions under

which the two features of the previous model (i.e. oxy-

gen-dependent peak formation of reduced PpsR and

bistability) persist if a more detailed mechanism for

the light regulation is employed. We are interested in

the potential kinetic constraints that the interplay

between light and redox regulation might impose on

the functionality of the AppA/PpsR system with a par-

ticular focus on the possibility of a bistable response.

We argue that the overexpression of AppA should

favour the experimental observation of a bistable

induction of PS genes. Finally, we show that our

model predictions can be brought into good agreement

with recent experimental results on the light-dependent

repression of PS genes under semi-aerobic conditions

[23,25]. To some extent, the model can also explain the

reduced blue-light sensitivity observed in an AppA

mutant, which contains a tryptophan 104 to phenylala-

nine base exchange in the FAD binding site of the

BLUF domain [25].

Model formulation

Below, we briefly summarize the main reaction steps

of the previous model, which essentially contains all

oxygen-dependent interactions between AppA and

PpsR. It also describes how the redox state of both

molecules depends on the ambient oxygen concentra-

tion. Subsequently, we extend this ‘redox module’ to

incorporate light-dependent effects. The full set of

oxygen- and light-dependent reaction steps is summa-

rized in Fig. 1.

Extended model of the AppA/PpsR signalling system R. Pandey et al.

2 FEBS Journal (2012) ª 2012 The Authors Journal compilation ª 2012 FEBS

Previous model: redox regulation

Both the light-inhibited complex formation between the

reduced forms of AppA (A)) and PpsR (P�4 ), as well as

the light-independent reduction of a disulfide bond in

oxidized PpsR (Pþ4 ) by the reduced form of AppA, were

modelled as reversible reactions of the form:

2A� þ P�4 ��! ��kþc =LI2

k�c

2AP2 ð1Þ

A� þ Pþ4 �! �kþPr

k�Pr

Aþ þ P�4 ð2Þ

where LI denotes light irradiance. Note that, in

Eqn (1), the light-inhibited complex formation between

AppA and PpsR is modelled in an effective manner.

For the benefit of a more detailed mechanism of light-

inhibition, the forward rate of this process will be

assumed to be independent of the light irradiance

when we discuss the extended model.

In Eqn (2), the parameters kþPr and k�Pr denote sec-

ond-order rate constants, and their ratio Keq ¼ kþPr=k�Pris related to the difference between the midpoint

potentials of the dithiol/disulfide couples in PpsR and

AppA via:

DEm ¼ EPþ4 =P�4m � EAþ=A�

m ¼ RT

2FlnKeq

At room temperature (298 K) the prefactor related to

the universal gas constant R and the Farady constant

F has a numerical value of RT/2F � 13 mV. Experi-

ments indicate that the electron transfer from AppA

to PpsR is effectively irreversible (i.e. Keq >> 1)

because, under a wide range of conditions, an inverse

electron flow from reduced PpsR to oxidized AppA

(A+) could not be observed [9]. This view is also sup-

ported by our simulations [24] revealing that particular

regulatory features, which we consider to be associated

with the phenotype of high light repression of PS

genes under semi-aerobic conditions, disappear as

Keq fi 1. For the sake of generality, we prefer to

model the electron transfer from AppA to PpsR as

reversible.

To implement the regulation of the redox states of

AppA and PpsR by the ambient oxygen concentration

we assumed simple first-order kinetics according to the

scheme [24]:

Aþ ��! ��kAr

kAo½O2�A� ð3Þ

Fig. 1. Scheme of the extended model for the blue-light- and oxygen-dependent interactions between AppA and PpsR as described by

Eqns (2–10). PpsR tetramers can exist either in an oxidized (Pþ4 ) or in a reduced state (P�4 ) where S–S and SH denote intramolecular disul-

fide bonds and thiol groups, respectively. Free AppA molecules can exist in four states corresponding to a reduced (A)) or an oxidized (A+)

heme cofactor and depending on whether the FAD cofactor is light-excited (A�� and Aþ� ). In addition, AppA and PpsR can reversibly associate

in a complex (AP2). Upon blue-light excitation, the light-excited complex (AP2)* irreversibly dissociates into A�� and half of a PpsR tetramer.

Numbers denote stoichiometric coefficients. Under aerobic conditions, PS genes are mainly repressed by the oxidized form of PpsR,

whereas the reduced form is the predominant repressor of PS genes under semi-aerobic conditions. The grey-shaded region encompasses

the light-dependent reactions. [O2] and LI denote the oxygen concentration and the light irradiance, respectively.

R. Pandey et al. Extended model of the AppA/PpsR signalling system


P�4 ��!kPo ½O2�Pþ4 ð4Þ

According to a model proposed by Han et al. [16]

AppA utilizes heme as a cofactor, bound to its C-ter-

minal domain, to sense the cytosolic redox conditions.

Consequently, A+ and A) correspond to an oxidized

and a reduced heme cofactor, respectively. In

Eqns (3, 4), we assumed that the reoxidation of AppA

and PpsR occurs in proportion to the concentration of

dissolved oxygen ([O2]) such that kAo[O2] and kPo[O2]

correspond to pseudo first-order rate constants,

whereas kAr denotes a first-order rate constant describ-

ing the reduction of AppA by an, as yet, unkown

mechanism.

Extended model: light regulation

Apart from being redox-active, AppA is a flavoprotein,

which enables it to act as a blue-light sensor by means

of a FAD cofactor bound to its N-terminal BLUF

domain [9,13,15]. Upon blue-light excitation of AppA,

the flavin undergoes a photocycle in the course of

which a long-lived signalling state is formed [9,26,27].

The transition to the signalling state is accompanied

by a conformational change in the AppA protein

[19,20], which is assumed to result in interactions with

its C-terminal part [19,16].

Based on these experimental results, it is unlikely

that light irradiation directly affects the association

rate between AppA and PpsR as assumed in Eqn (1).

Hence, in the extended model we assume that revers-

ible complex formation occurs independently of the

light irradiance (LI) as:

2A� þ P�4 �! �kþc

k�c2AP2 ð5Þ

where kþc and k�c denote an effective third- and a sec-

ond-order rate constant, respectively. Here, we only

account for the overall stoichiometry of this process,

although it is likely that is occurs in multiple steps.

Using a quasi-steady-state approximation, we have,

however, shown how the effective parameters kþc and

k�c in Eqn (5) can be related to the kinetic parameters

of an underlying multistep process [24].

To implement a more realistic model for the light-

dependent interaction between AppA and PpsR, we

follow the model proposed by Han et al. [16]. Accord-

ing to this model, light-induced structural changes of

AppA mediate the dissociation of the AppA–PpsR

complex and prevent the rebinding of PpsR to light-

excited AppA when the C-terminally bound heme is in

its reduced state (Fig. 1). Consequently, we introduce

three new states Aþ� , A�� and (AP2)* corresponding to

the light-excited forms of oxidized and reduced AppA,

respectively, as well as to the light-excited form of

AppA when bound in a complex with PpsR. Light reg-

ulation is modelled as a simple two-state process where

the excitation rate is proportional to the light irradi-

ance (LI) and the thermal recovery is described by a

first-order rate constant as:

A� ��! ��kþl�LI

k�l

A�� ð6Þ

Aþ ��! ��kþl�LI

k�l

Aþ� ð7Þ

AP2 ¢kþl�LI

k�l

AP2ð Þ��!kd

A�� þ1

2P�4 ð8Þ

In Eqn (8), kd denotes a first-order rate constant that

describes the light-induced dissociation of the AppA–

PpsR complex in an effective, yet stoichiometrically

correct, manner.

To complement the transitions between the four

AppA species, as defined in Eqns (3,6,7), we assume

that redox regulation of AppA occurs independently

of the light excitation:

Aþ� ��! ��kAr

kAo ½O2�A�� ð9Þ

Note that the rate constants in the cyclic reactions

shown in Eqns (3), (6), (7) and (9) are chosen in such

a way that ‘detailed balance’ holds.

Finally, we have to account for the observation that

the reduced form of AppA can reduce the disulfide

bond in oxidized PpsR irrespective of the light excita-

tion of the flavin [9], which leads to:

A�� þ Pþ4 �! �kPrþ

k�Pr

Aþ� P�4 ð10Þ

To keep the number of unknown parameters as low as

possible, we have assumed that the rates of reduction

and re-oxidation are the same as for the non-excited

forms of AppA (Eqn 2).

Rate equations and parameter definitions

By assuming mass-action kinetics the dynamics of the

reaction network in Eqns (2–10) is described by the

ordinary differential equations (ODEs):



d P�4� �dt¼� kPo O2½ � P�4

� �þ kþPr A�½ � þ A��

� �� Pþ4� �

þ kd2

AP2ð Þ��

� kþc A�½ �2 P�4� �

� k�c AP2½ �2� �

� k�Pr Aþ½ � þ Aþ��

P�4� �

d Pþ4½ �dt¼ kPo O2½ � P�4

� �� kþPr A�½ � þ A��

� �� Pþ4� �

þ k�Pr Aþ½ � þ Aþ��

P�4� �

d AP2½ �dt

¼ 2 kþc A�½ �2 P�4� �

� k�c AP2½ �2� �

� kþl LI½ � AP2½ �

þ k�l AP2ð Þ��

d AP2ð Þ��

dt¼ kþl LI½ � AP2½ � � k�l þ kd

� �AP2ð Þ�

� �d A�½ �

dt¼ kAr Aþ½ � � kAo O2½ � A�½ � � kþl LI½ � A�½ �

þ k�l A��

� 2 kþc A�½ �2 P�4� �

� k�c AP2½ �2� �

� kþPr A�½ � Pþ4� �

þ k�Pr Aþ½ � P�4� �

d A�� dt

¼ kþl LI½ � A�½ � � k�l A��

þ kAr Aþ��

� kAo O2½ � A��

� kþPr A��

Pþ4� �

þ k�Pr Aþ��

P�4� �

þ kd AP2ð Þ��

d Aþ�� dt

¼� kAr Aþ��

� kAo O2½ � A��

þ kþl LI½ � Aþ½ �

� k�l Aþ��

þ kþPr A��

Pþ4� �

� k�Pr Aþ��

P�4� �

d Aþ½ �dt¼� kAr Aþ½ � þ kAo O2½ � A�½ � � kþl LI½ � Aþ½ �

þ k�l Aþ��

þ kþPr A�½ � Pþ4� �

� k�Pr Aþ½ � P�4� �

ð11Þ

Here, we have assumed that the total amounts of the

proteins PpsR and AppA are conserved:

½Pþ4 � þ ½P�4 � þ1

2½AP2� þ

1

2½ðAP2Þ�� ¼ ½PT� and

½Aþ� þ ½A�� þ ½Aþ� � þ ½A�� þ ½AP2� þ ½ðAP2Þ�� ¼ ½AT� ð12Þ

where PT and AT denote the total concentrations of

PpsR and AppA, respectively. This assumption is in

agreement with the fact that the expression level of

PpsR were found to be largely independent of the

growth conditions [28]. However, the regulation of

AppA is not known. Hence, we will treat the ratio

c ¼ [AT]/[PT] as a free parameter in our study. To be

consistent with these assumptions we also neglect

dilution terms as a result of the cell growth in the

expressions in Eqn (11).

To analyze the steady-state behavior of the ODE

system in Eqn (11), we introduce dimensionless para-

meters (Table 1). This allows us to study the relative

significance of individual reaction steps for a certain

type of behaviour at the same time as keeping the

number of free parameters as small as possible. In

addition, concentrations are measured in terms of the

total protein concentrations as defined in Table 2. In

dimensionless units, the ODE system, thus, reads:

d

dsx1 ¼ x5�Ox1�

2dc

x21x2�Kc

x23

c2

� ��g Ix1� x6ð Þ

�bc

x1x4�x2x5

Keq

� �

d

dsx2 ¼ b x4 x1þ x6ð Þ� x2 x5þ x7ð Þ

Keq

� ��d x2

1x2�Kcx2

3

c2

� �

�aOx2þkg2

x8

d

dsx3 ¼ 2d x2

1x2�Kcx2

3

c2

� ��g Ix3� x8ð Þ

d

dsx6 ¼ x7�Ox6�

bc

x4x6�x2x7

Keq

� �þg Ix1� x6þ

kc

x8

� �

d

dsx7 ¼g Ix5� x7ð Þ� x7þOx6þ

bc

x4x6�x2x7

Keq

� �

d

dsx8 ¼g Ix3� x8�kx8ð Þ ð13Þ

where x4 and x5 are given by the dimensionless form

of the conservation relations (Eqn 12):

x4 ¼ 1� x2 �x3

2� x8

2and

x5 ¼ 1� x1 � x6 � x7 �x3

c� x8

c

In Eqn (13), time (s) is measured in units of 1/kAr.

The two main parameters of the present study are the

oxygen concentration ([O2]), which is measured in

units of K0 ¼ kAr/kAo, and the light irradiance (LI),

which is measured in units of KL ¼ k�l =kþl (typically

lmolÆm)2Æs)1). The parameters a, b, c and Keq are

defined in the same way as in our previous model [24],

whereas d is now independent of the light irradiance

(LI) as a result of the redefinition of kþc in Eqn (5).

Table 1. Definition of dimensionless parameters.

a ¼ kPo

kAo

b ¼ kþP r

AT½ �kAr

c ¼ AT½ �PT½ � d ¼ kþc ½AT �2

kAr

O ¼ ½O2 �KO

I ¼ LIKL

KO ¼ kAr

kAoKL ¼

k�l

kþl

g ¼ k�l

kAr

k ¼ kd

k�l

Keq ¼kþ

P r

k�P r

Kc ¼ k�ckþc PT½ �

Table 2. Definition of dimensionless state variables.

x1 ¼ ½A��½AT �

x2 ¼ ½P�4�

½PT � x3 ¼ ½AP2 �½PT � x4 ¼

½Pþ4�

½PT �

x5 ¼ ½Aþ�½AT � x6 ¼ ½A��

½AT � x7 ¼ ½Aþ� �½AT � x8 ¼ ½ðAP2Þ� �

½PT �



For direct comparison with the results of our previous

study, it is important to keep in mind that the defini-

tions of KL and d are different from those of the previ-

ous model. Another consequence of this redefinition is

that the effective dissociation constant Kc now plays

the same role as the square of the dimensionless light

irradiance (I2) in the previous model.

Finally, there are two new parameters:

g ¼ k�lkAr

and k ¼ kdk�l

which have a direct effect on light regulation. Specifi-

cally, g represents the ratio between the rates of the

thermal recovery of light-excited AppA species and

that of AppA reduction whereas k compares the

dissociation rate of the light-excited complex with the

thermal recovery rate.

Steady-state relations

Even though a detailed analytical steady-state analysis

of Eqn (13) is not feasible without making simplifying

assumptions, it is straightforward to derive some rela-

tions that show how the oxygen concentration and the

light irradiance determine the fraction of oxidzed ver-

sus reduced and light-excited versus non-excited states,

respectively, under steady-state conditions:

x6 þ x7 þx8

c¼ I

1þ Ið14Þ

x1 þ x5 þx3

c¼ 1

1þ Ið15Þ

x5 þ x7

x1 þ x6 þ ac x2¼ O ð16Þ

The first two equations describe how the light irradiance

determines the relative concentration of light-excited

(Eqn 14) and non-excited (Eqn 15) states. Similarly,

Eqn (16) shows that the oxygen concentration

determines the ratio between oxidized and reduced

states.

Results and Discussion

For convenience, we summarize the main features of

our previous model [24] below in advance of a discus-

sion of the results for the extended model. For most of

the simulations, we have assumed that the electron

transfer from AppA to PpsR in Eqn (2) is effectively

irreversible, as suggested by the experimental results of

Masuda and Bauer [9]. Note that this corresponds to

the limit Keq fi ¥ in Eqn (13) or k�Pr ¼ 0 in Eqn (2).

Peak formation and bistability in the previous

model

We have shown previously [24] that the steady-state

behavior of the AppA/PpsR system can be character-

ized in terms of only three parameters (a, b, c).Depending on their relative magnitude, we identified

two interesting types of behaviour: (a) the occurrence

of a maximum in the response curve of the reduced

form of PpsR at intermediate oxygen concentrations

under high light conditions and (b) the possibility of a

bistable regime in the transition from anaerobic to aer-

obic conditions (Fig. 2). Although the formation of a

peak at intermediate oxygen levels could provide an

explanation for the light-dependent PS gene repression

(mediated by the reduced form of PpsR) under semi-

aerobic conditions [21], experimental support for the

occurrence of bistability in the AppA/PpsR system is

still missing.

The most important parameter for the occurrence of

a maximum in the concentration of reduced PpsR (P�4 )

is b ¼ kþPr½AT�=kAr, which compares the time scale for

the reduction of AppA (1/kAr) with that for the reduc-

tion of PpsR (1=kþPr½AT�). If b is sufficiently large

A B

C D

Fig. 2. Peak formation and bistability in the previous model [24].

(A,B) At a ¼ 1 and c ¼ 2, decreasing Kc lowers the maximum of

reduced PpsR (½P�4 �) at intermediate oxygen concentrations

(O � 2), as well as the amount of free reduced PpsR (P�4 ) under

anaerobic conditions (O ¼ 0). (C,D) At a ¼ 10 and c ¼ 4, decreas-

ing Kc results in a bistable response. In the region between the

two saddle-node bifurcations (SN), two stable steady-states (solid

lines) coexist with one unstable steady-state (dashed line). Other

parameters: b ¼ 103 and Keq ¼ ¥.



(b >> 1), a pronounced maximum appears at interme-

diate oxygen concentrations. The position of this maxi-

mum along the oxygen axis can be adjusted by

changing the parameter a ¼ kPo/kAo (Fig. 2A,C),

which compares the timescale for the oxidation of

AppA (1/kAo[O2]) with that of PpsR (1/kPo[O2]).

Finally, the parameter c ¼ [AT]/[PT] represents the

ratio between total amounts of AppA ([AT]) and PpsR

([PT]). It determines the level of reduced PpsR under

anaerobic conditions. Because of the 2 : 1 stoichiome-

try of the reaction describing the complex formation

between AppA and PpsR (Eqn 5), we argued that

AppA copy numbers should exceed those of PpsR by

at least a factor of 2 (c ‡ 2) to allow for an efficient

sequestration of PpsR by AppA. If, in addition to

b >> 1, the requirement aO > c > 2 is fulfilled, the

AppA/PpsR system can exhibit a bistable response in

a certain parameter range.

If we set g ¼ 0 in Eqn (13) for the extended model,

it is straightforward to see that the total amount of

light-excited species is conserved: d(x5 + x6 + x7)/

dt ¼ 0. If we set the initial concentrations of these spe-

cies to zero, they will remain zero in time. Hence,

under these conditions, we recover the structure of our

previous model [24]. Specifically, as Kc is decreased,

the maximum of reduced PpsR at intermediate oxygen

concentrations (O � 2) is lowered (Fig. 2A, B).

Because decreasing the value of Kc increases the effec-

tive binding affinity for complex formation between

AppA and PpsR, the amount of free reduced PpsR

molecules is lower under anaerobic conditions (O ¼ 0).

Similarly, when a and c are chosen such that the con-

dition aO > c > 2 is fulfilled, lowering Kc can also

induce a bistable response in the transition from the

anaerobic to the aerobic regime (Fig. 2C, D).

Estimation of parameter values for the extended

model

The model, as defined by Eqn (13), contains 12 param-

eters (Table 1), two of which (KO and KL) can be

‘absorbed’ into the definition of the dimensionless oxy-

gen concentration (O ¼ [O2]/KO) and the dimensionless

light irradiance (I ¼ LI/KL), respectively. The parame-

ters KL, Kc and d can be estimated by combining

results from experimental measurements with general

mechanistic reasoning (Doc. S1). There, we find that

KL � 1 lmolÆm)2Æs)1, Kc > 0.1 and d ¼ 2,. . .,80. Rea-

sonable ranges for the parameters a, b, c and Keq are

suggested by the analysis of our previous model

(Fig. 2). A biologically plausible range for the two

remaining parameters, g ¼ k�l =kAr and k ¼ kd=k�l ,

can be estimated, where k�l describes the thermal relax-

ation of light-excited AppA back to the ground state.

Experiments have shown that, upon blue-light excita-

tion, AppA undergoes a photocycle in the course of

which a long-lived signalling state is formed [9,26,27].

The half-life of the signalling state was found to be

15 min [9] corresponding to k�l � 10�3 � s�1. On the

other hand, kAr characterizes the rate of reduction of

AppA, whereas kd is related to the light-induced con-

formational change of AppA. It can be expected that

both processes occur on a significantly faster time scale

compared to 1=k�l . For example, conformational

changes of proteins typically occur on a timescale of

milliseconds [29], such that we expect k >> 1 and

g>1.

Light-dependent peak formation in the extended

model

Recent in vivo experiments showed that the AppA/

PpsR system responds to blue-light signals down to a

light irradiance of 0.2 lmolÆm)2Æs)1 where half maxi-

mal repression of the puc gene was observed [23]. Satu-

rating levels of puc gene repression were reached at

LI � 1 lmolÆm)2Æs)1 and they remained constant up to

LI ¼ 20 lmolÆm)2Æs)1. This suggests that, in vivo, the

range LI ‡ 1 lmolÆm)2Æs)1 can already be regarded as

‘high light’ conditions, although in vitro studies often

used a light irradiance that was up to two orders of

magnitude larger [21,28,9]. For a light irradiance of

LI £ 0.1 lmolÆm)2Æs)1 no repression of PS genes were

observed, which thus marks the lower bound for the

sensitivity range with respect to light.

For a reasonable set of parameters (see above), the

extended model predicts the formation of a light-

dependent maximum in the concentration of reduced

PpsR at intermediate oxygen concentrations (Fig. 3).

Similar to our previous model [24], this peak forma-

tion requires a sufficiently large time scale separation

between the reduction rate of PpsR and that of AppA

(b >> 1). In addition, two trends are apparent. First,

at a fixed value of b, the difference between the maxi-

mum under low light conditions (0.1 lmolÆm)2Æs)1, nogene repression) and that under high light conditions

(1 lmolÆm)2Æs)1, maximal gene repression) is deter-

mined by the binding affinity (Kc) of the AppA-PpsR

complex. For smaller values of Kc, when complex for-

mation between AppA and PpsR occurs with higher

affinity, the difference between the maxima in the con-

centration of reduced PpsR gets larger. Second, peak

formation is more pronounced when the time scale

separation between AppA and PpsR reduction is large

(b ¼ 103), although it does not vanish under low light

conditions (LI ¼ 0.1, Fig. 3A, B). By contrast, when



b ¼ 102 (corresponding to an intermediate time scale

separation), the response curve of reduced PpsR still

develops a maximum at intermediate oxygen con-

centrations, although peak formation under low light

conditions is significantly suppressed, especially when

Kc ¼ 10)4 (Fig. 3C, D).

Unfortunately, the response curves in Fig. 3 cannot

be directly compared with experiments because mea-

surements of reduced PpsR under different light condi-

tions are not available. However, under the hypothesis

that the specific repression of PS genes under semi-aer-

obic conditions is associated with the occurrence of a

maximum in the response curve of reduced PpsR, we

expect the parameters b and Kc to be constrained by

two antagonistic goals. Peak formation apparently

requires a sufficiently large time scale separation

between AppA and PpsR reduction (b >> 1). On the

other hand, increasing b decreases the difference

between the maxima of reduced PpsR under high and

low light conditions. Consequently, the range of PpsR

concentrations over which downstream systems (speci-

fically PS gene transcription) would have to respond

becomes smaller, requiring a higher sensitivity of these

systems. Hence, to guarantee a proper functionality of

the AppA/PpsR system with respect to both light and

redox regulation, we expect b and Kc to be constrained

such that peak formation at intermediate oxygen levels

becomes possible (b sufficiently large) at the same time

as maintaining a sufficiently large difference of the

maximal PpsR concentrations between high and low

light conditions (b not ‘too large’ and Kc sufficiently

small).

Light regulation under semi-aerobic conditions

Two studies recently investigated the blue-light depen-

dent regulation of PS genes by the AppA/PpsR system

under semi-aerobic conditions in vivo [23,25]. These

studies showed that the most substantial changes in

the expression level of the puc gene occurred over

only one order of magnitude in the range of light

irradiance between 0.1 lmolÆm)2Æs)1 (no repression)

and 1 lmolÆm)2Æs)1 (maximal repression). Interestingly,

when puc gene repression reached maximal levels, the

relative repression level was only approximately 70%,

indicating a residual transcriptional activity even under

high light conditions.

Both features can be reproduced with the extended

model for the parameter set used in Fig. 3A. To com-

pare the response curve of reduced PpsR as a function

of the light irradiance (Fig. 4, solid curve) with experi-

mental measurements of puc gene inhibition (Fig. 4,

filled circles), we have assumed that the extent of puc

inhibition is proportional to the amount of reduced

PpsR bound to DNA. Using the results of DNA-bind-

ing experiments reported by Masuda and Bauer [9],

the relationship between puc inhibition and the frac-

tion of DNA-bound PpsR (x2 ¼ ½P�4 �=½PT�) can be

described by a simple Hill function of the form (Fig. 4,

dashed line):

puc inhibition ¼ 100xn2

Kn þ xn2,

where K ¼ EC50/[PT] and n denote a relative DNA-

binding affinity and an effective Hill coefficient, respec-

tively. The EC50 was found to be 69 nm [9], whereas

the effective Hill coefficient was estimated from

DNA-binding measurements as n � 3.4 (Fig. S1). Note

that, in Fig. 4, no attempt was made to find a set of

parameters that fits the experimental data points ‘opti-

mally’. Hence, we conclude that there exists a biologi-

cally reasonable set of parameters leading to a

response curve in agreement with the experimental

measurements. Our mathematical model can also be

used to analyze a phenotype with substantially dimin-

ished light sensitivity, as has been recently observed in

an AppA mutant strain [25] (Doc. S1 and Fig. S2).

Fig. 3. Light-dependent peak formation in the extended model

(Eqn 13). Shown are the steady-state response curves of reduced

PpsR (P�4 ) as functions of the oxygen concentration (O) for differ-

ent values of Kc and b. Small numbers denote the light irradiance

(LI) in lmolÆm)2Æs)1. (A, B) As Kc increases (from left to right), the

maximum of reduced PpsR (½P�4 �) at intermediate oxygen levels

(O � 2) increases under low light conditions (LI ¼ 0.1 lmolÆ-

m)2Æs)1). (C, D) As b decreases (from top to bottom), the maximum

of reduced PpsR at intermediate oxygen levels decreases signifi-

cantly under low light conditions. Parameters: a ¼ 1, c ¼ 2, Keq ¼¥, g ¼ 10)2, k ¼ 102, d ¼ 10.



To demonstrate how parameter changes affect the

shape of the response curve shown in Fig. 4 (solid

line), we have generated a set of response curves for

different combinations of the parameters d, k and g(Fig. 5A, B). In general, changing any of these param-

eters affects the steepness of the response curve and/or

the LI50 value (i.e. the light irradiance where half-

maximal levels of reduced PpsR are reached). For

example, increasing the parameter d ¼ kþc ½A2T�=kAr,

which describes the time scale separation between com-

plex formation and AppA reduction, increases the LI50at the same time as leaving the steepness almost

unchanged (Fig. 5A). However, decreasing d not only

leads to a smaller LI50, but also significantly weakens

the steepness of the response curve. Hence, we con-

clude that the time scales for complex formation and

AppA reduction have to be appropriately balanced to

generate a light response similar to that observed

experimentally.

A similar conclusion can be drawn with respect to

changes of the parameters k ¼ kd=k�l and g ¼ k�l =kAr,

both of which have a direct effect on light regulation

(Eqns 6–8). In section Estimation of Parameter Values

for the Extended Model we have argued that, as a

result of the small relaxation rate of the AppA signal-

ling state (k�l � 10�3 � s�1), we expect that k >> 1 and

g > 1. As shown in Fig. 5B, increasing the value of gsignificantly affects the steepness of the response curve

as well as the LI50, both of which are lowered. On the

other hand, a tenfold decrease of k to a value of 10

has almost no effect on the response curve. Only when

the relaxation rate k�l becomes comparable with the

dissociation rate of AppA–PpsR complexes (k ¼ 1) a

modest increase in the LI50 is observed. In summary,

these results suggest that a proper response to light sig-

nals in the experimentally observed range requires a

clear time scale separation between redox- and light-

dependent processes (g > 1) whereas the precise value

of k appears to be less important as long as k ‡ 1.

Figure 5C,D shows that the possible conclusions

are independent of the particular value of b, which

only causes a ‘vertical’ shift of the response curves,

whereas the effects of d, k and g on the response

behaviour remain qualitatively the same. This indi-

cates that the qualitative behaviour of the system

remains robust within an extended region of the

parameter space.

Effect of c ¼ [AT]/[PT] on the PpsR peak-position

and bistability

We have shown previously [24] that the ratio between

total copy numbers of AppA and PpsR (c ¼ [AT]/[PT])

not only determines how many PpsR molecules can be

sequestered by AppA under anaerobic conditions, but

Fig. 4. Comparison of the model prediction for the light response

with experiments. The solid line denotes the response curve of

reduced PpsR (P�4 ) for the parameter set Kc ¼ 10)4, a ¼ 1, b ¼103, c ¼ 2, d ¼ 10, Keq ¼ ¥, g ¼ 10)2, k ¼ 102, O ¼ 2 (Fig. 3A).

The dashed curve represents the percentage of puc gene inhibition

as calculated from: puc inhibition ¼ 100� xn2 =ðK n þ xn

2 Þ with

x2 ¼ ½P�4 �=½PT� (Table 2), n ¼ 3.4 (effective Hill coefficient) and

K ¼ EC50/[PT] ¼ 0.7 where EC50 denotes the concentration for half-

maximal saturation. Circles represent experimental measurements

(two repetitions) of puc inhibition taken from Metz et al. [25].

A B

C D

Fig. 5. Effect of d, k and g on the light response under semi-aero-

bic conditions (O ¼ 2). Solid lines represent the steady-state

response curves of reduced PpsR (P�4 ) as functions of the light irra-

diance (LI) for b ¼ 103 (A, B) and b ¼ 102 (C, D) for the parameter

set used in Fig. 4: a ¼ 1, c ¼ 2, d ¼ 10, k ¼ 102, g ¼ 10)2, KC ¼10)4, Keq ¼ ¥. Dashed lines indicate how the shape and/or the

position of the reference response curve would be affected by

independently changing one of the parameters d (A, C), k or g

(B, D) as indicated. Dotted lines mark the region of light irradiance

over which repression of the puc gene changes from its minimal

value (LI ¼ 0.1 lmolÆm)2Æs)1) to its maximal value (LI ¼ 1 lmolÆ

m)2Æs)1) as observed experimentally [23,25].



also whether or not bistability is possible. Specifically,

we have shown that an efficient sequestration requires

c ‡ 2, whereas bistability can occur for aO > c > 2.

Figure 6 shows that in the extended model, c also

affects the position of the maximum in the response

curve of reduced PpsR. Specifically, for c ¼ 1, the

maximum occurs at a fixed oxygen concentration

depending on the parameter a ¼ kPo/kAo but not on

the light irradiance (Fig. 6A,B). By contrast, as c is

increased, the position of the PpsR maximum is shifted

towards larger oxygen concentrations and a bistable

response becomes possible (Fig. 6C,D). This suggests

that, if a well-defined semi-aerobic regime is to exist

(i.e. light-dependent peak formation occurs in a nar-

row region of oxygen concentrations), then there

should exist a stoichiometric constraint for the ratio

between total amounts of AppA and PpsR (c ¼ [AT]/

[PT] must not become too large). Because PpsR expres-

sion levels ([PT]) were found to be largely independent

of the growth conditions [28], this means that AppA

expression levels ([AT]) would have to be regulated in

such a way that c £ 2 (Fig. 3). This indicates that

c � 2 might be an optimal value in the sense that it

allows for an efficient sequestration of PpsR at the

same time as minimizing the light-dependent shift of

the semi-aerobic regime (Fig. 6).

Figure 7 indicates that the occurrence of bistability

is restricted to low light conditions (I £ 1 correspond-

ing to LI £ 1 lmolÆm)2Æs)1) and intermediate oxygen

levels (Fig. 7A). Similar to our previous model [24],

the existence of bistability appears to require that aand b assume sufficiently large values (Fig. 7B,C),

which implies that there must be a sufficiently large

time scale separation between processes that modulate

the redox states of AppA and PpsR. By contrast, the

requirements for the parameters k and g, both of

which have a direct effect on light regulation, appear

to be much less stringent. Indeed, both parameters can

vary over several orders of magnitude within a broad

band without leaving the bistability region (Fig. 7D).

Although, as yet there is no experimental evidence for

a bistable response in the AppA/PpsR system, our

results suggest that a straightforward way to test this

possibility would be to overexpress AppA by providing

some extra copies of the appA gene [28]. In our model,

an increase of the total AppA concentration (AT) would

lead to an increase in the parameters b, c and d(Table 1). Because the region of bistability increases as

these parameters become larger (Fig. 7), increasing AT

should facilitate the observability of a bistable response.

To prevent interference with the PrrB/PrrA two-compo-

nent system, which specifically induces PS gene

expression under anaerobic conditions, it appears

A B

C D

Fig. 6. Light-dependent shift of the peak position and bistability.

Shown are the steady-state curves of reduced PpsR (P�4 ) as func-

tions of the oxygen concentration (O ) for different combinations of

a ¼ kPo/kAo and c ¼ [AT]/[PT]. Small numbers denote the light irradi-

ance LI in lmolÆm)2Æs)1. (A, B) When c ¼ 1, the maximum of

reduced PpsR at intermediate oxygen levels occurs at a fixed oxy-

gen concentration which depends on the value of a. (C, D) When c

is increased, the position of the maximum is shifted towards larger

oxygen concentrations in a light-dependent manner, and the

response becomes bistable. In the region between the two saddle-

node bifurcations (SN), two stable steady-states (solid lines) coexist

with one unstable steady-state (dashed line). Other parameters:

Kc ¼ 10)4, b ¼ 103, Keq ¼ ¥, g ¼ 10)2, k ¼ 102, d ¼ 10.

A B

C D

Fig. 7. Regions of bistability (shaded regions) projected on different

two-parameter planes. In the grey-shaded regions, two stable

steady-states coexist with one unstable steady-state. These regions

are bounded by two saddle-node (SN) bifurcations. Parameters

other than shown in the respective panel have the value: Kc ¼10)4, a ¼ 10, b ¼ 103, c ¼ 4, d ¼ 10, Keq ¼ ¥, g ¼ 10)2, k ¼ 102,

O ¼ 1, I ¼ 0.1.



advantageous to perform such experiments with a PrrB

knockout strain [30]. A heterogeneous response in the

expression of PS genes could be an advantageous sur-

vival strategy for a population of photosynthetic bacte-

ria aiming to cope with fluctuations in oxygen and light

availability, especially under semi-aerobic and low light

conditions.

Effect of Keq on peak formation

Similar to our previous model [24], peak formation at

intermediate oxygen levels is completely suppressed as

the equilibrium constant Keq for the electron transfer

between AppA and PpsR (Eqn 2) approaches 1 (Fig. 8).

Hence, we expect that the in vivo Keq is sufficiently large

(Keq >> 10), which is in agreement with the experimen-

tal observation by Masuda and Bauer [9] according

to which this electron transfer is effectively irreversible.

Summary and conclusions

We have proposed and analyzed a mathematical model

for the oxygen- and light-dependent interaction

between the AppA and PpsR proteins comprising part

of a signal transduction chain involved in the regula-

tion of PS genes in the facultative photosynthetic bac-

terium R. sphaeroides. Based on our previous model

[24], we have incorporated a more realistic mechanism

for the light-dependent interactions between AppA and

PpsR, as proposed by Han et al. [16]. Unlike the PrrB/

PrrA system, which is specifically involved in PS gene

induction under anaerobic conditions [18,1], the

AppA/PpsR system does not represent a standard two-

component system [31], although it also consists of a

sensory protein (AppA) that modulates the activity of

an associated effector protein (PpsR) in response to

environmental signals. Because AppA integrates both

blue-light and redox signals, R. sphaeroides exhibits a

specific phenotype under semi-aerobic conditions,

where PS genes are repressed by sufficiently strong

blue-light illumination [21,22,23]. Because an AppA-

homologue does not appear to exist in other purple

bacteria, this phenotype seems to be unique to R. sph-

aeroides [14].

With the help of the mathematical model, we have

analyzed the potential kinetic and stoichiometric

requirements for the regulatory processes between

AppA and PpsR that could explain the emergence of

such a phenotype. We have shown that, using biologi-

cally plausible parameter values, the model predictions

can be brought in congruence with experimental mea-

surements of light-dependent PS gene repression under

semi-aerobic conditions. Also, the model can qualita-

tively account for the reduced light sensitivity observed

in an AppA mutant strain [25]. Our results suggest

that the specific light-dependent repression of PS genes

at intermediate oxygen levels is caused by two time

scale separations in the AppA/PpsR interaction net-

work. The first time scale separation arises when the

rate of PpsR reduction is much larger than that of

AppA (i.e if b is sufficiently large). Under these condi-

tions, the steady-state curve for reduced PpsR exhibits

a pronounced maximum at intermediate oxygen con-

centrations and the height of this maximum decreases

in a light-dependent manner (Fig. 3). Depending on

the sensitivity of signal transduction systems down-

stream of PpsR, we expect b to lie in the range 100–

1000. The second time scale separation arises from the

fact that the AppA signalling state has a comparably

long half-life of approximately 15 min [9,25]. Our

results indicate that the corresponding relaxation rate

(k�l ) has to be much smaller than the rate of AppA

reduction (kAr) to ensure a proper response to light sig-

nals under semi-aerobic conditions (Figs 4 and 5).

In addition, we found that constraining the ratio

between total amounts of AppA and PpsR (c ¼ [AT]/

[PT]) to the range between 1 and 2 could help prevent

the occurrence of a significant light-dependent shift of

the semi-aerobic regime (Fig. 6) at the same time as

still allowing for an efficient sequestration of PpsR by

AppA. However, this would preclude the possibility of

a bistable response (Fig. 7). Our analysis suggests that

A B

C D

Fig. 8. Dependence of peak formation on the reversibility of the

electron transfer from AppA to PpsR (Eqn 2). As the equilibrium

constant Keq approaches unity, the maximum in the concentration

of reduced PpsR (P�4 ) at intermediate oxygen concentrations

(O � 2) vanishes. Numbers denote light irradiance LI in lmolÆ-

m)2Æs)1. Parameters are the same as shown in Fig. 3A: Kc ¼ 10)4,

a ¼ 1, b ¼ 103, c ¼ 2, d ¼ 10, g ¼ 10)2, k ¼ 102.



a simple way to induce or favour a bistable response is

to increase the total concentration of AppA relative to

that of PpsR (e.g. by overexpressing the appA gene).

Materials and methods

The steady-state response curves shown in Figs 2–6 and 8

were generated with the simulation packages xppaut [32]

and matcont [33]. Model ODE files are available from the

authors upon request.

Acknowledgements

R.P. and R.S. acknowledge financial support from the

‘International Max Planck Research School Magde-

burg’ and from the Research Center ‘Dynamic Systems’

funded by the Ministry of Education of Saxony-Anhalt,

respectively.

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Supporting information

The following supplementary material is available:

Fig. S1. Estimation of the Hill coefficient from experi-

mental data.

Fig. S2. Comparison of the model predictions with

experiments in the AppA mutant strain W104F.

Doc. S1. Estimation of KL, Kc and d and modelling

the light response in the AppA mutant strain W104F.

This supplementary material can be found in the

online version of this article.

Please note: As a service to our authors and readers,

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may be re-organized for online delivery, but are not

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should be addressed to the authors.



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