the role of positive feedback loops involving anti-dsdna and anti–anti-dsdna antibodies in...

Journal of Theoretical Biology 319 (2013) 8–22

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Journal of Theoretical Biology

0022-51

http://d

n Corr

Cambrid

E-m

journal homepage: www.elsevier.com/locate/yjtbi

The role of positive feedback loops involving anti-dsDNAand anti–anti-dsDNA antibodies in autoimmune glomerulonephritis

A. Arazi a,b,n, A.U. Neumann a,b

a Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan, Israelb Institute for Theoretical Biology, Humboldt University, Berlin, Germany

H I G H L I G H T S

c Anti–anti-dsDNA Abs explain the persistence of anti-dsDNA Abs in disease remission.c Their occurrence does not change the parameter sensitivity of IC-mediated disease.c These Abs explain why the knockout of TLR9 can lead to an exacerbated inflammation.c Limit cycles account for reported inverse oscillations of the two antibody types.

a r t i c l e i n f o

Article history:

Received 19 November 2011

Received in revised form

9 September 2012

Accepted 17 September 2012Available online 8 November 2012

Keywords:

Autoimmune glomerulonephritis

Systemic lupus erythematosus

Autoimmune inflammation

Anti-idiotypic antibodies

TLR9

93/$ - see front matter & 2012 Published by

x.doi.org/10.1016/j.jtbi.2012.09.017

esponding author. Present address: Broad Ins

ge Center, Cambridge, MA 02142, USA. Tel.:

ail address: [email protected] (A. Arazi).

a b s t r a c t

Autoimmune glomerulonephritis (GN) is a potentially life-threatening renal inflammation occurring in

a significant percentage of systemic lupus erythematosus (SLE) patients. It has been suggested that GN

develops and persists due to a positive feedback loop, in which inflammation is promoted by the

deposition in the kidney of immune complexes (IC) containing double-stranded DNA (dsDNA) and

autoantibodies specific to it, leading to cellular death, additional release to circulation of dsDNA,

continuous activation of dsDNA-specific autoreactive B cells and further formation of IC. We have

recently presented a generic model exploring the dynamics of IC-mediated autoimmune inflammatory

diseases, applicable also to GN. Here we extend this model by incorporating into it a specific B cell

response targeting anti-dsDNA antibodies—a phenomenon whose occurrence in SLE patients is

well-supported empirically. We show that this model retains the main results found for the original

model studied, particularly with regard to the sensitivity of the steady state properties to changes in

parameter values, while capturing some disease-specific observations found in GN patients which are

unaccountable using our previous model. In particular, the extended model explains the findings that

this inflammation can be ameliorated by treatment without lowering the level of anti-dsDNA

antibodies. Moreover, it can account for the inverse oscillations of anti-dsDNA and anti–anti-dsDNA

antibodies, previously reported in lupus patients. Finally, it can be used to suggest a possible

explanation to the so-called regulatory role of TLR9, found in murine models of lupus; i.e., the fact

that the knockdown of this DNA-sensing receptor leads, as expected, to a decrease in the level of anti-

dsDNA antibodies, but at the same time results in a counter-intuitive amplification of the autoreactive

immune response and an exacerbated inflammation. Several predictions can be derived from the

analysis of the presented model, allowing its experimental verification.

& 2012 Published by Elsevier Ltd.

1. Introduction

Autoimmune disorders are often suggested to be the result ofcomplex processes, involving multiple superimposed non-lineareffects (Beutler and Bazzoni, 1998; Christensen and Shlomchik,2007; Jang et al., 2009; Kroemer et al., 1988; Ogura et al., 2008;Shlomchik et al., 2001). One such example is a certain class of

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autoimmune diseases which involve a self-sustaining inflamma-tion resulting from the deposition of immune complexes (IC) inthe inflamed tissue. The persistence of this type of inflammationis due to a positive feedback loop, where autoantigen particlesreleased as part of the tissue damage caused by the inflammationlead to the activation of autoreactive B cells, the secretion ofautoantibodies, the formation of further immune complexes andtheir subsequent deposition (Fig. 1a). A number of autoimmuneinflammatory disorders are thought to be the result of suchprocesses, including autoimmune glomerulonephritis in sys-temic lupus erythematosus patients (Clynes et al., 1998;

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Fig. 1. Modeling autoimmune glomerulonephritis as a result of two positive feedback loops. (a) The basic positive feedback loop suggested to underlie autoimmune

inflammations mediated by the deposition of immune complexes, studied previously (Arazi and Neumann, 2010). Autoantigen particles (here, dsDNA; denoted by A)

released to circulation following tissue damage and cell death stimulate specific autoreactive B cells (B1), leading to the secretion of autoantibodies. Immune complexes

(I) formed by the interaction of these autoantibodies and the autoantigen particles are deposited in the tissue, promoting further inflammation and cell death by activating

local effector cells. Note that the initial tissue damage responsible for triggering this feedback loop may be external to it, e.g., cell death due to an invading pathogen.

(b) The model studied in the present paper, incorporating a second positive feedback loop implied in autoimmune glomerulonephritis, in which the anti-dsDNA

autoantibodies stimulate anti-idiotypic B cells (B2) specific to them. Upon activation, these B cells proliferate and secrete antibodies which by themselves can stimulate the

dsDNA-specific B cells. (c) The curve describing the effective proliferation rate of B cell population i, as a function of the stimulation signal sensed by these cells, assumed to

be proportional to the concentration of the corresponding antigen/antibody to which the cells are specific. In accordance with both empirical and theoretical results, it is

assumed that this curve has a log bell shape. Li and Hi are the values of the stimulation signal hi for which pBif iðhiÞ ¼ dBi , i.e., the effective proliferation rate of population i is

equal to its death rate. The horizontal axis is given in a logarithmic scale.

A. Arazi, A.U. Neumann / Journal of Theoretical Biology 319 (2013) 8–22 9

Schiffer et al., 2002), rheumatoid arthritis (De Clerck, 1995),idiopathic pulmonary fibrosis (Chapman et al., 1984; Dall’Aglioet al., 1988) and polyarteritis nodosa (Herbert and Russo, 2003;Reumaux et al., 2004).

We have recently presented a generic model exploring thedynamics of IC-mediated autoimmune inflammatory diseases(Arazi and Neumann, 2010). Using it, we were able to predictthe identity of the major factors governing the development ofsuch pathologies. In particular, we showed that the existenceand stability of steady states representing disease depend, to alarge extent, on the relation between three parameters: themaximal rate of autoantigen release to circulation, the clear-ance rate of immune complexes, and the affinity betweenimmune complexes and their binding sites in the tissue. Thisis in contrast to the emphasis given by some investigators tothe pathogenic role played by deficient clearance of circulatingautoantigen (Munoz et al., 2005; Walport, 2000). These findingsmay serve to guide the development of future therapies to suchinflammations.

In the present work we use this model to further study onespecific IC-mediated autoimmune inflammatory disorder, namelyautoimmune glomerulonephritis (GN). This is a severe kidneyinflammation occurring in a significant percentage of systemiclupus erythematosus (SLE) patients, which may lead to end-stagerenal failure and death. In this disease, the main autoantigenthought to drive the above positive feedback loop is double-stranded DNA (dsDNA). Here we extend our original model by

introducing into it a second population of B cells that specificallyreact with the anti-dsDNA antibodies. These anti-idiotypic B cellssecrete antibodies which interact with the anti-dsDNA antibodies,and which are further capable of stimulating the dsDNA-specificautoreactive B cells. We emphasize that we do not assume here inany way the existence of an idiotypic network, relating multiple Bcell clones—a concept found to be highly questionable; rather, wetake into account only the (first-level) B cell response targetingautoantibodies, whose existence is well-founded empirically inlupus, as well as several other diseases. Indeed, the presence ofanti–anti-dsDNA antibodies in both SLE patients (Abdou et al.,1981; Zhang et al., 2001) and animal models of this disease (Eilatet al., 1985; Migliorini et al., 1987) has long been established;moreover, their binding activity to anti-dsDNA antibodies wasshown to correlate with disease remission, suggesting that theyplay a role in the modulation of disease (Abdou et al., 1981;Williams et al., 1995; Zouali and Eyquem, 1983). Furthermore, itwas experimentally demonstrated that anti–anti-dsDNA antibo-dies can stimulate dsDNA-specific B cells, such that the injectionof these antibodies into mice led to an increase in the level ofanti-dsDNA antibodies and to induction of disease (Mendlovicet al., 1989); i.e., the stimulatory relationship between the two Bcell populations was indeed shown to be bi-directional. We thusadd to the original model a second positive feedback loop, relatingthese two cell populations (Fig. 1b).

The resulting model is studied analytically. We show that itcan be used to suggest explanations to several observations found

A. Arazi, A.U. Neumann / Journal of Theoretical Biology 319 (2013) 8–2210

specifically in GN patients, while retaining the main properties ofour original generic model, and in particular the above-mentioned relative importance of the different model parameters.Several predictions are derived, making the theoretical workpresented here experimentally verifiable.

2. Model

The original model we investigated (Arazi and Neumann,2010) describes the positive feedback loop through whichIC-mediated autoimmune inflammations are suggested to sustainthemselves (Fig. 1a). It is defined by the equations

dA

dt¼ smax

A

I

yAþ I�dAA�kIAB1,

dI

dt¼ kIAB1�dII,

dB1

dt¼ sB1þpB1f 1 h1ð ÞB1�dB1B1: ð1Þ

Table 1lists the variables and parameters in this model, as well astheir units. Briefly, A denotes the concentration of circulatingautoantigen; I is the concentration of circulating immune com-plexes, formed from autoantigen particles and the autoantibodiestargeting them; and B1 designates the total number of autoreac-tive B cells, specific to the autoantigen. A single variable is used torepresent all the different life stages of these cells (i.e., naı̈ve cells,activated cells, plasmablasts, etc.). The level of antibodies isassumed to be roughly proportional to the level of the B cellpopulation producing them—an assumption which is kept in thepresent work as well; the implications of its relaxation will bestudied elsewhere.

Autoantigen particles are released to circulation due to thetissue damage promoted by the deposition of immune complexes,and therefore the rate of this process is taken to be a rising andsaturating function of I, with a maximal value equal to smax

A ; thisterm can be derived from a simplified model of the deposition andinflammation processes, assuming these are fast enough com-pared to the other processes described in the above equations(Arazi and Neumann, 2010). The clearance of circulating auto-antigen particles is assumed to occur in a constant rate dA. kI is theeffective rate of immune complexes formation, taking intoaccount the affinity of the autoantibodies, their valence and thevalence of the autoantigen molecules, and the proportionalityfactor assumed to exist between the levels of the autoreactive

Table 1The meaning and units of the model variables and parameters. For es

(2010).

Variable/parameter Meaning

A Circulating autoantigen (dsDNA) conce

I Immune complexes (IC) concentration

B1 Total number of dsDNA-specific B cells

B2 Total number of anti-idiotypic B cells

smaxA Maximal rate of autoantigen release d

yA IC concentration yielding half the max

dA Autoantigen clearance rate

kI Effective rate of immune complexes fo

dI Immune complexes clearance rate

sBi Production rate of B cell population i ð

pBi Maximal proliferation rate of B cell po

yB1i Activation threshold of B cell populatio

yB2i Suppression threshold of B cell popula

dBi Death rate of B cell population i

aA Factor translating the stimulation per

B cells and the antibodies they produce. The clearance rate ofcirculating immune complexes is denoted by dI . The autoreactiveB cells are assumed to be generated in the bone marrow with aconstant rate sB1, and to die with a constant rate dB1. Theireffective proliferation rate – pB1f 1ðh1Þ – is a function of thestimulation signal sensed by the B cells, h1; in the original model,it was assumed that h1 is proportional to A. Several alternativefunctional forms for f 1ðh1Þ were considered in the above work;below we focus on one of them.

We have shown that the behavior of this system is controlled,to a large extent, by a certain composite parameter, denoted by B0

and given by the expression

B0 ¼dAdIyA

kIðsmaxA �dIyAÞ

: ð2Þ

B0 is the lowest B1 value on the non-trivial nullcline of A; i.e., it isthe minimal level of autoreactive B cells required for a steadystate featuring a self-sustained inflammation. Clearly, it is moresensitive to changes in dI , the clearance rate of immune com-plexes, than to changes in dA, the clearance rate of circulatingautoantigen—an observation with direct clinical implications.These findings hold even when one alters the functional form ofseveral of the terms in Eq. (1).

In the present work, we focus on a certain disease – auto-immune glomerulonephritis – in which the main autoantigenfeeding the above feedback loop is thought to be dsDNA. Themodel defined by Eq. (1) is extended here by incorporating into itan additional population of B cells, whose level is denoted by B2,which are stimulated by the anti-dsDNA autoantibodies, andwhich secrete antibodies that stimulate, in turn, the autoreactive,dsDNA-specific B cells (Fig. 1b). Accordingly, we set h1 ¼ aAAþB2,h2 ¼ B1; that is, in the extended model the dsDNA-specific B cellsare assumed to respond (to different degrees) to both thecirculating dsDNA and the anti–anti-dsDNA antibodies. Thedynamics of the anti-idiotypic B cell population are given by theequation

dB2

dt¼ sB2þpB2f 2ðh2ÞB2�dB2B2, ð3Þ

The effective proliferation rate of population i (i¼1,2) is takento be a log bell-shaped function of the antigen concentration(Fig. 1c), i.e., it is initially an increasing function, but decreases forextremely high concentrations of antigen—an assumption sup-ported both empirically (Celada, 1971) and theoretically (Perelson

timates of some of the parameter values, see Arazi and Neumann

Units

ntration pM

pM

cell

cell

ue to tissue damage pMd�1

imal rate of autoantigen release pM

d�1

rmation cell�1d�1

d�1

i¼ 1,2Þ cell d�1

pulation i d�1

n i cell

tion i cell

d�1

autoantigen Molar to stimulation per B cell cell pM�1


and DeLisi, 1980). Here we set, following De Boer and Hogeweg(1988)

f iðhiÞ ¼hi

yB1iþhi�

yB2i

yB2iþhi: ð4Þ

The parameters yB1i and yB2i may be thought of as the activationand suppression thresholds of population i, respectively. It isassumed that yB2ibyBi1; in this case, pBi approximates the max-imal effective proliferation rate. The two values of hi in whichpBif iðhiÞ ¼ dBi, denoted here by Li and Hi (Fig. 1c), are of importancein the analysis detailed below; it can be shown thatLi � ðdBi=ðpBi�dBiÞÞyB1i, Hi � ððpBi�dBiÞ=dBiÞyB2i. Another value ofinterest is VBi � sBi=dBi, the basal level of population i, found inthe absence of stimulation.

We further assume a quasi-steady state with regard to I, anassumption readily justified by the relatively high clearance rateof immune complexes, even in lupus patients (Davies, 1996;Kimberly et al., 1986). This allows us to write I as a product ofA and B1, and thus to work with a 3-dimensional model, which westudy analytically (a non-dimensional version of this model isgiven in Appendix A; however, since the number of parameters init is still quite high, and since we wish to focus on the effect ofparameters having a direct biological meaning, we do not furtherinvestigate it).

Fig. 3. The steady states in the A¼0 plane. The nullclines of B1 are shown in green,

those of B2 in blue (thicker lines). Overall, there are five steady states in this plane.

Potentially stable steady states are designated by filled circles; unstable steady

states are denoted by empty ones. (For interpretation of the references to color in

this figure caption, the reader is referred to the web version of this article.)

3. Results

3.1. Steady states

Setting the equations for A, B1 and B2 to 0 defines a set ofsurfaces, depicted in Fig. 2, whose intersection points constitutethe steady states of the model. In particular, from the equation forA, one can see that there are two sets of steady states in this

Fig. 2. The surfaces on which the model equations are equal to 0. (a) The surface resultin

shown). (b) The surfaces corresponding to the equation for B2. (c)–(d) The surfaces cor

this and the following figures, all axes are presented in a logarithmic scale, in order to p

To allow the schematic representation of the A¼0 plane on a logarithmic scale, an arbitr

in the generation of all figures.

system: those residing in the A¼0 plane, and those in which A40(throughout this manuscript, X denotes the steady state value ofvariable X). The former are free of circulating autoantigen andtherefore of inflammation mediated by deposited immune com-plexes. Five steady states belong to this set (Fig. 3): in the first ofthese, B1 � VB1, B2 � VB2, i.e., both B cell populations are found intheir basal levels, in the absence of any effective stimulation. Inthe other four steady states B1 is roughly equal to L2 or H2, and B2

is approximated by L1 or H1: in these steady states the two B cellclones balance each other, but without any pathologic involve-ment. All five steady states exist, given the reasonable assumptionthat VB1oL2 and VB2oL1; below we denote them by sVV, sLL, sLH,sHL and sHH (see Fig. 3).

g from equating the equation for A to 0. To this one should add the plane A¼0 (not

responding to the equation for B1 (split to two panels for readability purposes). In

oint out the geometrical properties of the surfaces corresponding to each variable.

arily small A value was used instead. See Appendix D for the parameter values used


The second set consists of steady states parallel to the above,in which A40, and where it is h1 ¼ aAAþB2, rather than B2 alone,which is approximately equal to L1 or H1; we refer to these as SVV,SLL, SLH, SHL and SHH (Fig. 4). In these steady states, the positivefeedback loop responsible for the persistence of an immunecomplex-mediated inflammation is active; therefore, they canbe thought of as disease states. The existence of the disease steadystates is parameter-dependent; a full specification of the relevantconditions is given in Appendix B. Below we show that three ofthem are potentially stable (that is, stable under certain condi-tions on parameter values)—the steady states in whichðB1,aAAþB2Þ are approximately equal to ðVB1,VB2Þ, ðL2,H1Þ orðH2,L1Þ (i.e., SVV, SLH and SHL). SVV may be thought of as represent-ing a mild disease, as it does not entail an effective activation andproliferation of the two B cell populations, unlike the other twosteady states, which correspond to a full-blown disease.

The disease steady states result from intersections with thesurface appearing in Fig. 2a, defined by the equation

A¼B1ðsmax

A �kIy0AÞ�dAy0AB1ðdAþkIB1Þ

� gðB1Þ, ð5Þ

where y0A ¼ ðdI=kIÞyA. This surface crosses the A¼0 plane at a B1

value equal to B0, defined by Eq. (2). Note that this is the same B0

as in the original model studied in Arazi and Neumann (2010),since the equation for A does not change in the extended model,and in particular does not depend on B2. There, we have shownthat this size is central in determining the behavior of the system;in the following section we show that this is true for the modelinvestigated in the present work as well. In particular, theexistence of the steady states in which A40, as well as the localstability of both sets of steady states, depend on the value of B0, ascompared to a series of thresholds.

We note that certain configurations of parameter values cangive rise to two additional stable disease states. The first of thesecorresponds to a case where a full-blown inflammation, involvingan extremely high level of anti-dsDNA antibodies, suppresses theanti-idiotypic response, since B1bH2; the system then degener-ates to the situation described by the original model (Fig. 4g). Theother additional steady state represents a case where the systemis over-flooded with circulating dsDNA, leading to the suppres-sion of the dsDNA-specific B cells, since AbH1, and therefore to alack of activation of the anti-idiotypic B cells (Fig. 4b).

3.2. Stability analysis

In this section we show that B0 indeed controls the stabilityand existence of the steady states of the system. In particular, weprove that if B0 is smaller than a value roughly equal to VB1, thesteady state SVV is locally stable, while its counterpart sVV isunstable. If we now change some of the parameter values andincrease B0, then as it crosses this threshold, these two steadystates merge, and the resulting steady state, in which A ¼ 0, isstable (Fig. 5a, b). Similarly, as long as B0 is smaller than a valueapproximated by L2, the steady state SLH is stable, while itsparallel in the A¼0 plane, sLH, is unstable. When B0 reaches thisthreshold, the two steady states merge into a steady state inwhich A ¼ 0, that is then locally stable (Fig. 5c, d). Finally, asimilar result holds also with respect to the steady states SHL andsHL, with a threshold approximately equal to H2 (Fig. 5e, f).

Define

FðA,B1,B2Þ ¼dA

dt¼ smax

A

AB1

y0AþAB1�dAA�kIAB1,

GðA,B1,B2Þ ¼dB1

dt¼ sB1þpB1f 1 h1ð ÞB1�dB1B1,

HðA,B1,B2Þ ¼dB2

dt¼ sB2þpB2f 2ðh2ÞB2�dB2B2: ð6Þ

The characteristic polynomial of the system is given by PðlÞ ¼l3þa1l

2þa2lþa3, where

a1 ¼�@F

@A�@G

@B1�@H

@B2,

a2 ¼@F

@A

@G

@B1þ@F

@A

@H

@B2þ@G

@B1

@H

@B2�@H

@B1

@G

@B2�@G

@A

@F

@B1,

a3 ¼@H

@B1

@G

@B2

@F

@Aþ@H

@B2

@G

@A

@F

@B1�@F

@A

@G

@B1

@H

@B2: ð7Þ

Here we have taken into account that @H=@A¼ 0, @F=@B2 ¼ 0.According to the Routh–Hurwitz conditions (Murray, 2003), asteady state is locally stable iff a140, a340 and a1a2�a340.

We begin with the steady states in which A ¼ 0. Here we canfurther rely on the fact that @F=@B1 ¼ 0. For sVV we have f 1ðh1Þ ¼

f 2ðh2Þ � 0, and so GðA,B1,B2Þ � sB1�dB1B1, HðA,B1,B2Þ � sB2�dB2B2.Thus, the condition a140 can be written as @F=@AodB1þdB2.Substituting A ¼ 0,B1 ¼ VB1 in @F=@A gives

@F

@A¼ VB1dA

smaxA �y

0

AkI

dAy0

A

�dA, ð8Þ

leading to the condition (from Eq. (2))

B04dA

dB1þdB2þdAVB1: ð9Þ

For the steady states sLL, sLH, sHL and sHH, we have pB1f 1ðh1Þ �

dB1,pB2f 2ðh2Þ � dB2, and so @G=@B1 � 0,@H=@B2 � 0. Thus, for thesesteady states, the condition a140 reads approximately as@F=@Ao0. When B1 � L2, this can be rewritten in a manneranalogous to Eq. (9) as

B04L2, ð10Þ

and, when B1 �H2, the equivalent requirement is

B04H2: ð11Þ

Relying on the fact that @F=@B1 ¼ 0 when A ¼ 0, the conditiona340 can be rewritten as

@F

@A

@G

@B2

@H

@B1�@G

@B1

@H

@B2

� �40: ð12Þ

For sVV we have @G=@B1 ��dB1o0,@H=@B2 ��dB2o0, @G=@B2 � 0,@H=@B1 � 0. Therefore, the expression in the parentheses is nega-tive. This dictates the condition @F=@Ao0, which is stricter than (9),and translates into

B04VB1: ð13Þ

For sLH we can use the fact that B15y22, B2by11, andapproximate GðA,B1,B2Þ � sB1þB1½pB1y21=ðy21þB2Þ�dB1�, HðA,B1,B2Þ �

sB2þB2½pB2B1=ðy12þB1Þ�dB2�. This yields @G=@B2o0, @[email protected], for this steady state, ð@G=@B2Þð@H=@B1Þo0. Similarly,the inequality ð@G=@B2Þð@H=@B1Þo0 also holds for sHL, but not forsLL and sHH. Since, as explained above, for these four steady stateswe have @G=@B1 � 0,@H=@B2 � 0, and since we already require forthem that @F=@Ao0, we see that the first two steady states fulfillthe condition a340, but not the latter ones, which are thereforenot stable.

We now turn to the condition a1a2�a340. After some rear-rangements, this reads as

a1a2�a3 ¼�@F

@A

� �2 @G

@B1þ@H

@B2

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

I

�@G

@B1

� �2 @F

@Aþ@H

@B2


II

�@H

@B2

� �2 @F

@Aþ@G

@B1


III

�2@F

@A

@G

@B1

@H

@B2|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}IV

Fig. 4. The steady states in which A40. Filled circles mark locally stable steady states, while empty circles refer to unstable steady states. AX is the value of gðB1Þ (Eq. (5))

in which B1 is equal to X.


Fig. 5. B0 controls the existence and local stability of the system’s steady states. As parameter values are changed and B0 increases, the disease steady states converge to

their counterparts in the A¼0 plane; merging occurs through bifurcation points, once B0 reaches certain thresholds (see text). Depicted are the three steady state pairs

which are potentially stable—sVV & SVV (panels a–b), sLH & SLH (c–d) and sHL & SHL (e–f). The left panels demonstrate the movement in space of each disease state as B0

increases—red arrows mark the change in B0, black arrows delineate the resulting shift in the position of the steady state. Stable steady states are denoted by filled circles,

while unstable ones by empty circles. The right panels show the change in the steady state level of circulating dsDNA (A), as function of B0; here the value of B0 is modified

by changing smaxA (changing B0 through other parameters yields similar results). Stable steady states are designated by a solid line, unstable states by a dashed line. These

panels were prepared using the XPP-AUTO software package (Ermentrout, 2002). Note that for SLH, in sufficiently low B0 values (i.e., sufficiently high A values), the disease

state may lose its local stability and a limit cycle may emerge through a Hopf bifurcation; this is shown in panel (d). (For interpretation of the references to color in this

figure caption, the reader is referred to the web version of this article.)


þ@H

@B1

@G

@B2

@G

@B1þ@H

@B2

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

V

40: ð14Þ

Note that I, II, III and IV all include the preceding minus signs.As shown above, for sLH and sHL we have @G=@B1 � 0,@H=@B2 � 0,where in fact these derivatives are close to 0 but are strictlynegative, since in the steady states hi is either slightly lower thanLi or slightly higher than Hi, due to the term designating theconstant influx of B cells from the bone marrow; for sVV we have@G=@B1 ��dB1o0,@H=@B2 ��dB2o0. In addition, we havealready required above that @F=@Ao0 for these three steadystates. Therefore, they all fulfill I,II,III,IV40. Furthermore, since@G=@B1þ@H=@B2o0, and since @G=@B2 � 0,@H=@B1 � 0 for sVV and

ð@G=@B2Þð@H=@B1Þo0 for the other two steady states, VZ0. Thus,these three steady states fulfill also the condition a1a2�a340.

In summary, we have proven that sufficient and necessaryconditions for the local stability of sVV, sLH and sHL are B04VB1,B04L2 and B04H2, respectively, where it should be noted thatthe rhs in each condition is an approximation of the actualthreshold. Furthermore, sLL and sHH are not stable.

Moving on to the steady states in which A40, we first considerthose in which both B cell populations proliferate, i.e., SLL, SLH, SHL

and SHH. Again, and for identical reasons as above, the conditiona140 is roughly equivalent to the requirement @F=@Ao0, i.e.,

smaxA

y0AB1

ðy0AþAB1Þ2�dA�kIB1o0: ð15Þ


The coordinates of these steady states satisfy Eq. (5). Therefore,we can write

y0AþB1 � A ¼ y0AþB1 �B1ðsmax

A �kIy0

AÞ�dAy0

A

B1ðdAþkIB1Þ¼

B1smaxA

dAþkIB1

: ð16Þ

Substituting in (15), we get the condition

@F

@A¼ ðdAþkIB1Þ

y0AðdAþkIB1Þ

smaxA B1

�1

" #o0: ð17Þ

This inequality holds iff the expression in the brackets is negative,i.e., iff

B0oB1: ð18Þ

Thus, in steady states where B1 � L2, the requirement is approxi-mately B0oL2; and in steady states where B1 �H2, it is B0oH2.

Turning again to Eq. (5), dividing and multiplying the numera-tor by dAy

0

A, we get

A ¼

dAy0

A

B1

B0�1

!

B1ðdAþkIB1Þ: ð19Þ

We can see that as long as B0oB1, A40, and the necessarycondition (18) holds, so that the steady state can bestable (depending on the other two conditions, see below). WhenB0 ¼ B1, we have from Eq. (19) A ¼ 0, and the steady state inwhich A40 merges with its counterpart in which A ¼ 0. Accordingto the results found above, the resulting steady state in the A¼0plane becomes stable in the cases where ðh1,h2Þ � ðL1,H2Þ orðh1,h2Þ � ðH1,L2Þ. This scheme is depicted in Fig. 5.

The condition a340 can be rewritten as

@F

@A

@G

@B2

@H

@B1�@G

@B1

@H

@B2


I

þ@H

@B2

@G

@A

@F

@B1|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}II

40: ð20Þ

First, from considerations similar to those used in checking thecondition a340 for the steady states in which A ¼ 0, we can seethat I40 for SLH and SHL, but not for SLL and SHH. Furthermore, inAppendix C we show that close enough to the A¼0 plane (and tothe bifurcation points) II is either positive or else small enough inits absolute value so as not to change the overall sign of a3. Forlarge A values in the case of SLH, or for intermediate A values inthe case of SHL, this condition may be violated, with implicationsdiscussed in Section 3.4.

We now turn to check the condition a1a2�a340. For the steadystates in which A40 we can write, after some rearrangement

a1a2�a3 ¼�@F

@A

� �2 @G

@B1þ@H

@B2

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

I

�@G

@B1

� �2 @F

@Aþ@H

@B2


II

�@H

@B2

� �2 @F

@Aþ@G

@B1


III

�2@F

@A

@G

@B1

@H

@B2|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}IV

þ@H

@B1

@G

@B2

@G

@B1þ@H

@B2


V

�@G

@A

@F

@B1

@F

@Aþ@G

@B1


VI

: ð21Þ

Since @G=@B1o0, @H=@B2o0, and since we already require@F=@Ao0, we have I,II,III,IV40. Furthermore, since for SLH andSHL we have ð@G=@B2Þð@H=@B1Þo0, it holds that V40. Finally, aslong as we are close to the A¼0 plane, @F=@B1 remains small in itsabsolute value (see Appendix C), and the condition a1a2�a34 0holds.

To summarize, we have shown that SLH is stable iff B0oL2 andif A is not too large; SHL is stable iff B0oH2, with the possible

exception of intermediate A values; and SLL and SHH are unstable,at least in the vicinity of the A¼0 plane. Again, it should be notedthat the rhs in the conditions referring to B0 are approximations ofthe actual thresholds.

In the steady states where B2 � VB2, i.e., when the anti-idiotypic clone is inactive, the system is approximately the sameas the one studied in Arazi and Neumann (2010) (note that inthese cases, either h25L2 or h2bH2, and therefore 9dB2=dt9remains small, i.e., B2 is more or less constant). The stabilityproperties of these steady states are thus identical to thosededuced for the model studied in that previous work.

3.3. Explaining the persistence of anti-dsDNA antibodies during

amelioration of disease

Experimental studies in murine models of SLE have shown thatit is possible to reduce the severity of an ongoing kidneyinflammation while keeping the high level of anti-dsDNA anti-bodies unchanged (Werwitzke et al., 2005; Bahjat et al.,2008)—an observation which cannot be accounted for by themodel we originally studied (Arazi and Neumann, 2010). In thatmodel, the nullcline of B1 is given by the equationB1 ¼ sB1=ðdB1�pB1f ðh1ÞÞ, and thus in the disease steady statesA � L1=aA (Fig. 6). As a result, any kind of therapy (assumed toaffect the value of some model parameter(s)) will necessarily leadto a change in B1 : A treatment that modifies the parametersgoverning the dynamics of autoantigen or immune complexeswill lead to a shift in the position of the nullcline of A, and thechange in the coordinates of the disease state will be such that itis rather the level of autoantigen that persists, while the level ofanti-dsDNA antibodies decreases (Fig. 6a). A treatment affectingthe parameters pertaining to the dsDNA-specific B cells will leadto a decrease in the level of anti-dsDNA antibodies and to anincrease in the level of circulating autoantigen (Fig. 6b).

In contrast, the persistence of anti-dsDNA antibodies followingtreatment is readily explained by the present model, at least inthe case of a therapy affecting parameters related to the dynamicsof the autoantigen or immune complexes (for example, it wassuggested that the modality used by Werwitzke et al. (2005)enhanced the clearance of immune complexes). Such a treatmentwill lead to an increase in the value of B0; the resulting shift in theposition of the steady states will keep the value of B1 roughlyunchanged, while decreasing A (as illustrated in Fig. 5, panels c & e).This is due to the fact that in a steady state involving theactivation of both B cell populations, B1 is equal to a level keepingthe anti-idiotypic B cells in equilibrium, i.e., approximately L2 orH2; that is, it does not depend on the parameters associated withthe concentrations of circulating dsDNA or immune complexes.

We note that a treatment affecting these parameters will alsoentail an increase in the steady state level of the anti-idiotypic Bcells B2 (due to the decrease in A), and therefore also in the levelof anti–anti-dsDNA antibodies—a prediction which can bechecked experimentally.

3.4. Occurrence of inverse oscillations in the levels of anti-dsDNA

and anti-idiotypic antibodies

The analysis of the stability properties of the disease states SLH

and SHL shows that local stability is maintained as long as thevalue of A is not too large. Numerical exploration reveals that ifthis condition is not fulfilled, a limit cycle may emerge from eachsteady state through a Hopf bifurcation (Figs. 5d and 7). Thisresult is consistent with the inverse oscillations in the levels ofanti-dsDNA and anti-idiotypic antibodies reported to occur insome lupus patients (Zouali and Eyquem, 1983). Such oscillations

Fig. 6. The effect of treatment in the original model. The original model, lacking the anti-idiotypic response, cannot account for the persistence of anti-dsDNA antibodies

following treatment ameliorating the kidney inflammation. (a) The effect of treatment modifying the parameters appearing in the equations of A or I. (b) The consequences

of treatment affecting the parameters appearing in the B1 equation. Dashed lines indicate the nullclines position following treatment; arrows delineate the change in the

phase plane position of the disease steady state.

Fig. 7. Examples of limit cycles in the studied model. (a) A limit cycle resulting from a Hopf bifurcation of the SLH disease state. (b) A limit cycle resulting from a Hopf

bifurcation of the SHL disease state.


cannot be explained by the model studied in Arazi and Neumann(2010) where, if the disease state exists, it is always locally stable.

3.5. Explaining the regulatory role of TLR9

It was previously demonstrated in murine models of SLE thatthe knockout of TLR9 – one of the main receptors responsible forDNA-sensing – results, as expected, in lower levels of circulatinganti-dsDNA antibodies but, at the same time, also leads toincreased lymphadenopathy, the accumulation of activated lym-phocytes, increased levels of total (non-anti-dsDNA) circulatingIgG antibodies and an exacerbated kidney disease (Christensenet al., 2006; Christensen and Shlomchik, 2007). This effect wasfound also when the knockout of TLR9 was performed in B cellsalone (and in particular not in dendritic cells). Here we show thatthe present model suggests a possible explanation to this counter-intuitive result.

TLR9 enhances the activation and differentiation of dsDNA-specific B cells, once these internalize circulating dsDNA interact-ing with their BCR (Christensen and Shlomchik, 2007). Therefore,one way to represent its knockout in the present model is toincrease the activation and suppression thresholds of these B cells(the parameters yB11 and yB21, respectively). We note that this is amore accurate representation of TLR9 knockout, compared toincreasing only the parameter aA, which represents the degree ofstimulation of the dsDNA-specific B cells by dsDNA molecules(in contrast to the stimulation these cells experience due to anti–anti-dsDNA antibodies): this is because following its initial liga-tion to internalized dsDNA, TLR9 enhances the response of thesecells to any antigen their BCR interacts with.

The shift in activation and suppression thresholds of thedsDNA-specific B cells allows the anti-idiotypic B cells to increasefaster and to a higher level on the expense of the formerpopulation, biasing the system towards the SLH disease staterather than the SHL disease state, which may be normally realized(Fig. 8, panels a, b). As a result, the level of anti-dsDNA antibodiessignificantly decreases (Fig. 8c), in accordance with experimentalresults, while the level of the anti–anti-dsDNA antibodies demon-strates a substantial increase. In general, the steady state level ofB2 is determined by the activation and suppression thresholds ofthe dsDNA-specific B cells: B2 is equal to a value such that in theequation for B1, the proliferation term (which is a function of B2

and the above thresholds) and the cell death term approximatelybalance each other, taking into account also the negligible effectof the term representing the influx of cells from the bone marrow.Therefore, if the shift in thresholds, and in particular in yB21, islarge enough, the increase in B2 will be sufficient to ensure thatthe new steady state is characterized by an overall higher level ofIgG antibodies, corresponding to the sum B1þB2, as compared tothe case prior to the knockout of TLR9 (Fig. 8d), in agreement withexperimental results. We note that, depending on parametervalues, it is possible that SLH does not even exist before theknockout of TLR9 (Fig. 9).

Thus, the present model lends a possible explanation to thehigher level of total circulating IgG antibodies (not necessarilyanti-dsDNA Abs) found in TLR9-knockout mice, with a predictionthat most of these antibodies will be anti–anti-dsDNA antibo-dies. However, the new steady state will inevitably feature adecrease in tissue damage, as shown in Fig. 8e: the reason forthis is that the steady state level of circulating immune

Fig. 8. The effect of knocking-out TLR9. The knockout of the TLR9 receptor, which enhances the activation and differentiation of the dsDNA-specific B cells, is represented

in the model by increasing the activation and suppression thresholds of these cells (yB11 and yB21, respectively). This may affect the competition between the dsDNA-

specific and the anti-idiotypic B cells, allowing the latter cell population to rise faster and to higher levels on the expense of the former one. Thus, while in the presence of a

functional TLR9, in the wild-type (WT) strain, the system tends to converge to the SHL disease state (panel (a)), following the knockout of TLR9 the system may be biased

towards the SLH disease state (panel (b)). Panels (c)–(f) show the steady state levels of several variables, both in the presence of a functional TLR9 (blue bars) and following

its knockout (red bars). (c) The level of dsDNA-specific B cells (B1), corresponding to the level of anti-dsDNA antibodies. In accordance with experimental results, the

knockout of TLR9 leads to lower levels of these autoantibodies. (d) The total level of dsDNA-specific and anti-idiotypic B cells (i.e., B1þB2), corresponding to the overall

level of IgG antibodies in the model, including antibodies other than anti-dsDNA antibodies. Again in accordance with experimental results, the knockout of TLR9 leads to a

counter-intuitive increase in this level. (e) The rate of autoantigen release (equal to smaxA I=ðyAþ IÞ), assumed to be proportional to the overall tissue damage caused by the

inflammation. In contrast with experimental results, this rate cannot increase in the present model following the knockout of TLR9; see text for ways to settle this

discrepancy. (f) The concentration of circulating dsDNA (A). The model predicts that the knockout of TLR9 will lead to a drastic increase in the steady state level of this

variable. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 9. The emergence of the SLH disease state following the knockout of TLR9. Depending on parameter values, it is possible that in the presence of a functional TLR9

(i.e., in the wild-type strain of the lupus-prone mice), the system does not initially feature a steady state in which B1 � L2 ,aAAþB2 �H1 (i.e., SLH), but rather a steady state

in which AbH1=aA ,B1 � VB1 ,B2 � VB2 (panel (a)). As explained in the text, the knocking-out of TLR9 is represented by an increase in the activation and suppression

thresholds of the dsDNA-specific B cells (yB11 and yB21, respectively), and therefore the surfaces along which the equation for B1 is equal to 0 (colored green in the figure)

‘‘move outwards’’ in the A�B2 plane (black arrows in panel (a)). As a result, the position of the intersection points of these surfaces with the surfaces corresponding to A

(red) and B2 (blue) changes. In particular, the coordinates of the above steady state following the change in these thresholds will be such that B1 � L2 ,aAAþB2 �H1 (panel

(b)). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)


complexes, I , is given by

I ¼kI

dIAB1 ¼

kI

dI

B1ðsmaxA �kIy0AÞ�dAy0AðdAþkIB1Þ

: ð22Þ

Therefore

@I

@B1

¼kI

dI

smaxA dA

ðdAþkIB1Þ240: ð23Þ


Hence, if we move from one steady state to another in which B1 issmaller, without changing the parameters appearing in the aboveequations (as is the case here, since we represent the knockout ofTLR9 by changing parameters appearing in the equation for B1

only), I will decrease. Consequently, the rate of autoantigenrelease to circulation, given by smax

A I=ðyAþ IÞ and assumed to beproportional to the intensity of the tissue inflammation, willnecessarily decrease. In other words, the new steady state reachedfollowing the knockout of TLR9 will not feature a more severeinflammation, in contrast with the empirical observation. This istrue also for the model studied in Arazi and Neumann (2010),which at any rate cannot explain the rise in the total level of IgGs.

There are at least two possible ways to further account for theexacerbated inflammation as well. First, a persisting high level ofcirculating anti-idiotypic antibodies may eventually affect the clear-ance rate of the pathogenic, dsDNA-containing immune complexes.A high level of anti–anti-dsDNA antibodies will lead to a high level ofimmune complexes composed of anti-dsDNA antibodies and anti-idiotypic antibodies, and thus to a possible saturation of the reticu-loendothelial system, which handles the clearance of immune com-plexes in general, with a potential spillover of immune complexesinto the kidney (Bradfield, 1974). Furthermore, high levels of circulat-ing antibodies may saturate the FcRn pathway, responsible forantibody recycling (Tabrizi et al., 2006), which was recently demon-strated to be involved also in preventing the accumulation of immunecomplexes in the glomerular basement membrane (Akilesh et al.,2008). Either way, a decrease in the efficacy of the mechanismsresponsible for the clearance of pathogenic immune complexes,represented here by reducing the value of dI , is expected. In the

Fig. 10. The effect of knocking-out TLR9, in a model incorporating the interactions bet

(Eqs. (1) and (3)), the knockout of TLR9 in the modified model (Eq. (24)) affects the com

than a convergence to SHL (panel (a)), the system will tend towards the SLH disease state

dsDNA antibodies (panel (c)) and the counter-intuitive increase in the overall level IgG

modified model can also account for the reported increase in tissue damage due to in

autoantigen release (the numbers on the bars designate the actual values in each case)

knockout of TLR9 will lead to a decrease in the steady state level of circulating dsDNA

reader is referred to the web version of this article.)

present model, this will indeed result in a rise of I and therefore in amore severe kidney inflammation.

Another possible explanation to the exacerbated kidney dis-ease may lie in the observation that anti–anti-dsDNA antibodiescan themselves also interact with DNA molecules (Eivazova et al.,2000; Fischel and Eilat, 1992), in accordance with the epibodyphenomenon, described by Bona et al. (1986). To represent this,we modify the model equations and write

dA

dt¼ smax

A

I

yAþ I�dAA�kI1AB1�kI2AB2,

dI

dt¼ kI1AB1þkI2AB2�dII,

dB1

dt¼ sB1þpB1f 1ðh1ÞB1�dB1B1,

dB2

dt¼ sB2þpB2f 2ðh2ÞB2�dB2B2: ð24Þ

This model differs from the one studied above by the fact thathere immune complexes are formed also from the interactionsbetween dsDNA and anti–anti-dsDNA antibodies. In addition,while the stimulation signal sensed by the dsDNA-specific B cells(B1) continues to be h1 ¼ aA1AþB2 that of the anti-idiotypic B cells(B2) now depends also on the level of dsDNA, i.e., h2 ¼ aA2AþB1.We assume that the main difference between the two B cellspopulations lies in the affinity of these interactions: i.e., weassume that kI2okI1, aA2oaA1.

We do not present here a full investigation of this model.However, numerical simulations show that here as well there are(at least) two disease steady states, corresponding to SHL (Fig. 10a)

ween anti–anti-dsDNA antibodies and dsDNA. As in the main model studied here

petition between the dsDNA-specific and the anti-idiotypic B cells, so that rather

(panel (b)). Both models account for the decrease in the steady state level of anti-

antibodies (panel (d)), found experimentally. However, unlike the main model, the

flammation, by demonstrating an increase in the steady state level of the rate of

(panel (e)). Finally, in contrast with the main model, here the prediction is that the

(panel (f)). (For interpretation of the references to color in this figure caption, the


and SLH (Fig. 10b). We note first that here, the knockout of TLR9 isrepresented by increasing the activation and suppression thresholdsof not only B1, but also those of B2, as the anti-idiotypic B cells alsointernalize circulating dsDNA, and as a result are stimulated by itsinteraction with TLR9. As in the model studied above, knocking-outTLR9 leads to a bias towards the steady state corresponding to SLH,and entails a decrease in the level of anti-dsDNA antibodies(Fig. 10c) and a counter-intuitive concurrent increase in the totallevel of IgG antibodies (Fig. 10d), in accordance with experimentalresults. Furthermore, this knockout gives rise to an increase in therate of autoantigen release (Fig. 10e), i.e., an increase in the tissuedamage caused by the autoimmune IC-mediated inflammation (oneshould note that while this change is not in orders of magnitude, it isstill substantial—50%, in the simulations shown here).

To decide between these two alternative explanations, one cansimply measure the steady state level of circulating dsDNA, A,following the knockout of TLR9. According to the main modelstudied here, a decrease in B1 will necessarily involve an increasein A, due to the shape of the surface on which the equation for A isequal to 0 (Figs. 2a, 8f). In contrast, in the model incorporating theinteractions between anti–anti-dsDNA antibodies and dsDNA, thesteady state reached after knocking-out TLR9 will involve adecrease in A (Fig. 10f).

4. Discussion

In this work, we modeled the pathogenesis and persistence ofautoimmune glomerulonephritis, a renal inflammation occurring in asubstantial percentage of SLE patients. Our model assumed theexistence of two positive feedback loops—one pertaining to theself-amplification of inflammation caused by the deposition ofimmune complexes, the other describing the bidirectional interac-tions between B cells producing anti-dsDNA and anti–anti-dsDNAantibodies. The occurrence of both the first (Clynes et al., 1998;Schiffer et al., 2002) and second (Abdou et al., 1981; Mendlovic et al.,1989; Zouali and Eyquem, 1983) feedback loops is well-foundedempirically. Through an analytical exploration of our model, weidentified two potentially stable steady states describing a full-blown disease—one featuring a high level of anti-dsDNA antibodies,the other a high level of circulating dsDNA and anti–anti-dsDNAantibodies; a third steady state, designating mild disease, was alsodemonstrated. The existence and local stability of these steady stateswere shown to depend on the relation between B0, a function of theparameters associated with the dynamics of autoantigen and immunecomplexes, and a series of thresholds; interestingly, these pertain tothe activation and suppression thresholds of the anti-idiotypic B cells,rather than the dsDNA-specific B cells. We showed that this modelcan explain the unexpected persistence of high levels of anti-dsDNAantibodies following the administration of a treatment amelioratinginflammation (Werwitzke et al., 2005; Bahjat et al., 2008). We furtherdemonstrated the existence of two limit cycles in the studied model,in accordance with the inverse oscillations in the levels of anti-dsDNAand anti–anti-dsDNA antibodies measured in some SLE patients(Zouali and Eyquem, 1983). Finally, we used our model to suggest apossible explanation to the so-called regulatory role found to beplayed by TLR9 in murine models of lupus, i.e., the counter-intuitiveobservation that the knockout in mice of TLR9 – one of the maincellular receptors sensing DNA – leads to a decrease in the levelof anti-dsDNA antibodies, as expected, but at the same time also toan increase in the overall level of IgG antibodies and to a moresevere kidney disease (Christensen et al., 2006; Christensen andShlomchik, 2007).

These three results – the persistence of anti-dsDNA antibodies,their periodic oscillations, and the effect of TLR9-knockout –cannot be obtained using the generic model we recently presented

(Arazi and Neumann, 2010), which consisted only of the firstpositive feedback loop mentioned above. Thus, the novel modelstudied here possesses an enhanced explanatory power. None-theless, we showed that most of the main properties of the genericmodel are retained here as well. In particular, the same compositeparameter, B0, is the major factor controlling the behavior of bothmodels. As a consequence, the clinical implications found for thegeneric model and stemming from the functional form of B0 holdhere as well: that is, treatments aiming to enhance the clearanceof immune complexes (i.e., increasing dI), or to mask them fromtheir binding sites in the tissue (increasing yA), are predicted to befar more effective than those trying to increase the clearance rateof cellular debris (dA). This result can guide the development offuture therapies to autoimmune glomerulonephritis.

One should note that the present model does not predict that all

treatments of autoimmune glomerulonephritis will preserve the levelof anti-dsDNA antibodies. Treatments which affect the parametersassociated with the anti-idiotypic population will result in a changein the steady state level of the anti-dsDNA antibodies, as theB1-coordinate in equilibrium is mainly determined by the surfaceson which _B2 ¼ 0. For example, anti-BLyS monoclonal antibodies(Ding, 2008), which suppress the activation of B cells in a non-BCR-specific manner, are expected to yield such a result.

The notion that the knockout of TLR9 affects the competitionbetween different B cell populations, allowing the rise of B cellswhich are not specific to dsDNA, was already suggested by others(Christensen and Shlomchik, 2007). The picture portrayed here is,however, more elaborate: as explained in Section 3.5, because ofthe bidirectional idiotypic interactions existing between the twoB cell populations considered, the knockout of TLR9 not onlyenables one population to rise on the expense of the other, butalso determines its new steady state level. That is, the larger theshift in the activation and suppression thresholds of the dsDNA-specific B cells, the higher the overall level of IgG antibodies willbe. This argument is, in principle, experimentally verifiable: if onecan produce quantitatively different inhibitions of the TLR9signaling pathway (e.g., by using different dosages of shRNAsspecific to proteins participating in this cascade), one can proceedto estimate yB21 in each case, by generating the antigen responsecurve of the dsDNA-specific B cells, and then to check if thisparameter correlates with the total level of IgG antibodies observed.

Simple idiotypic networks composed only of anti-Id and anti–anti-Id B cell populations were studied extensively in the past (De Boeret al., 1990, 1993a,b; Perelson, 1989; Stewart and Varela, 1990).Autoimmunity in the context of anti-idiotypic immune responses wasinvestigated in a number of works. Weisbuch et al. (1993) have usedan idiotypic network model to suggest an explanation to theprevalence of autoimmune diseases involving hormones and theirreceptors. Borghans et al. (1998) have explored the properties of T cellvaccinations by investigating a system composed of autoreactiveT cells and anti-idiotypic regulatory T cells. The multiplicity of auto-antibodies in systemic lupus erythematosus was suggested to be theresult of a percolation in the idiotypic network, promoted by theoccurrence of loops such as the one stemming from the interactionbetween anti–anti-dsDNA antibodies and dsDNA (Neumann andWeisbuch, 1992). However, none of these works dealt with inflam-mations promoted by immune complex deposition.

Here we suggested the idiotypic interactions, known to indeedexist in the studied system, as means of explaining the persis-tence of anti-dsDNA antibodies following treatment. We note thatother explanations may be proposed for this observation. Forexample, one may argue that the dsDNA-specific B cells remaincontinuously activated due to co-stimulatory signals receivedfrom bystander cells, i.e., non-specific cells activated during theperiod of inflammation. However, such an explanation is ques-tionable, as the level of the ‘‘danger signal’’ in the system (and in


particular the level of proinflammatory cytokines and othermolecules secreted by such bystander cells) is expected todecrease with the amelioration of inflammation, thus leading toa decrease also in the level of anti-dsDNA antibodies—in contrastto the actual observation. Another alternative explanation is thatlong-lived plasma cells continue to secrete anti-dsDNA antibodieseven after the activation of naı̈ve dsDNA-specific B cells hasceased; indeed, long-lived plasma cells were found in murinemodels of SLE (Hoyer et al., 2004). In future work, we will explorethis possibility. Other directions in which the current model maybe extended include the incorporation of antibody dynamics andthe addition of other populations of immune cells.

The model studied here is only slightly more complicated thanthe model we previously discussed (Arazi and Neumann, 2010),resulting from adding a single biological component in order toexplain a certain observation unaccounted for; as a byproduct, wewere able to identify the potential underlying causes of additionalempirical phenomena. This reflects a more general approach tomodel building: The almost endless complexity of biologicalsystems warrants the ultimate usage of models far more detailedthan the one used here, if one wishes to explain every knownphenomenon. However, there is no clear procedure instructingone in deciding which biological details should be included insuch a model. A possible solution is to start with a minimal model(as was done in our previous work), and to map the observationsthat can be explained by such a model, as well as those thatcannot. The latter can point to possible incremental steps ofmaking the model more realistic, while trying to keep it as simpleas possible. In such a way, one may hope to identify a minimal set

of biological components and interactions required to explain awidest possible range of phenomena. Additional complications ofour model are, without a doubt, required; however, their inves-tigation is beyond the scope of the present manuscript.

While the work described here is strictly theoretical – mainly dueto the lack of relevant detailed published data – it is nonethelessexperimentally verifiable. Indeed, several predictions were derivedhere. First, a rise in the level of anti–anti-dsDNA antibodies ispredicted to accompany treatments in which the amelioration ofkidney disease is associated with a persistence of anti-dsDNA anti-bodies. Furthermore, we predict that the mechanism of action ofthese treatments is such that it leaves unaffected the parametersassociated with the dynamics of the anti-idiotypic B cells, given therole of these parameters in determining the B1-coordinate in thesteady states of the system, as explained above. With regard to TLR9-knockout mice, we predict that most of the circulating IgG antibodieswill be anti–anti-dsDNA antibodies. Furthermore, we predict that inthese mice either the clearance rate of immune complexes decreaseswith time, or the anti–anti-dsDNA antibodies interact also withdsDNA, and suggest measuring the steady state level of circulatingdsDNA as means of deciding between these two alternatives. Thus,our findings can be, in principle, experimentally validated.

Acknowledgements

A. Arazi is funded by the Converging Technologies Ph.D.scholarship, given by the Israeli Council for Higher Education,and by the Bar-Ilan University President’s Ph.D. scholarship.

Appendix A. A non-dimensional model

A non-dimensional version of the model is given by theequations

dA

dt¼

I

1þ I�a1A�a2AB1,

dI

dt¼ AB1�I,

dB1

dt¼ a3þa4

a5AþB2

a6þa5AþB2

a7

a7þa5AþB2B1�a8B1,

dB2

dt¼ a9þa10

B1

a11þB1

a12

a12þB1B2�a13B2: ðA:1Þ

where all the variables and parameters are dimensionless, and

a1 ¼ dA=dI , a2 ¼ yAdI=smaxA , a3 ¼ sB1kIsmax

A =d3I yA, a4 ¼ pB1=dI , a5 ¼

aAkIðsmaxA Þ

2=d3I yA, a6 ¼ yB11kIsmax

A =d2I yA, a7 ¼ yB21kIsmax

A =d2I yA, a8 ¼

dB1=dI , a9 ¼ sB2kIsmaxA =d3

I yA, a10 ¼ pB2=dI , a11 ¼ yB12kIsmaxA =d2

I yA,

a12 ¼ yB22kIsmaxA =d2

I yA, a13 ¼ dB2=dI . The number of free para-

meters is thus reduced from 16 in the original model to 13.

Appendix B. A full list of the disease steady states and theconditions for their existence

Define Amax as the maximal A value on the surface given byEq. (5), and AX as the value of gðB1Þ in which B1 is equal to X. Thenthe steady states in which A40, and the conditions for theirexistence, are as follows (Fig. 4):

1.
L1=aAbA40,B1 � VB1,B2 � VB2 (Fig. 4a; referred to as SVV inthe text). Necessary conditions for the existence of this steadystate are B0oVB1 and AL2
oL1=aA. This steady state can bethought of as representing a mild inflammation, not entailing afull-blown activation and proliferation of the two B cellpopulations. With this said, for parameter values in which A

approaches L1=aA, this steady state involves some increase inthe level of dsDNA-specific B cells. When A eventually reachesa threshold approximately equal to L1=aA, this steady statemerges with the steady state SLL through a saddle-nodebifurcation.

2.
AbH1=aA,B1 � VB1,B2 � VB2 (Fig. 4b). Necessary conditions forthe existence of this steady state are B0oL2 and Amax4H1=aA.In this steady state, the system is over-flooded with circulatingautoantigen, leading to the suppression of the dsDNA-specificB cells, and therefore there is no activation of the anti-idiotypicB cells.
3.
A40,B1 � L2,aAAþB2 � L1 (SLL; Fig. 4c). Necessary conditionsfor the existence of this steady state are B0oL2 andAL2
oL1=aA. This steady state, as well as the next 3, can bethought of as representing a full-blown disease.

4.
A40,B1 � L2,aAAþB2 �H1 (SLH; Fig. 4d). Necessary conditionsfor the existence of this steady state are B0oL2 andAL2
oH1=aA. When AL2approaches H1=aA, this steady state

eventually merges with the steady state in whichAbH1=aA,B1 � VB1,B2 � VB2.

5.
A40,B1 �H2,aAAþB2 � L1 (SHL; Fig. 4e). A sufficient conditionfor the existence of this steady state is B0oH2. Note that ifAH2
ZL1=aA, this steady state becomes a steady state whereA � ðL1�VB2Þ=aA,B1bH2,B2 � VB2 (Fig. 4g), i.e., the system isover-flooded with anti-dsDNA antibodies, and the anti-idiotypic response is suppressed. Furthermore, if B04H2 andAmax4L1=aA, there are two steady states involving this phe-nomenon (of which only the one with a higher B1 coordinate islocally stable).

6.
A40,B1 �H2,aAAþB2 �H1 (SHH; Fig. 4f). A sufficient condi-tion for the existence of this steady state is B0oH2. IfAH2
ZH1=aA, the coordinates of this steady state becomeA � ðH1�VB2Þ=aA,B1bH2,B2 � VB2 (Fig. 4h), so here too, anextremely high level of anti-dsDNA antibodies leads to thesuppression of the anti-idiotypic response. Again, if B04H2


and Amax4H1=aA, there are two steady states with suchcoordinates.

These observations are made simply by considering all thepossible intersections between the surfaces presented in Fig. 2.For example, if B04L2, the surface defined by the equation forA lies entirely above the lower surface defined by equating theequation for B2 to 0 (see Fig. 2b), and thus steady states in whichB1rL2 do not exist.

Appendix C. Showing that a340 for the steady states SLH

and SHL

In Section 3.2, in trying to prove that a3 ¼ IþII40, we haveshown that I40 for SLH and SHL but not for SLL and SHH. Of course,the condition a340 may still be violated for the former twosteady states if II is ‘‘negative enough’’. We note, first, that @H=@B2

is small but strictly negative. Furthermore, the sign of @G=@A

depends on the specific steady state studied (see below). We turnto investigate the behavior of @F=@B1, which is given by

@F

@B1¼ smax

A

y0AA

ðy0AþAB1Þ2�kIA: ðC:1Þ

When B0 is equal to the threshold corresponding to the steadystate, A ¼ 0 and therefore @F=@B1 ¼ 0, and so II¼ 0. If we nowslightly modify the value of one of the parameters so as todecrease B0 (e.g., by increasing smax

A ), A starts to increase. Forsmall A values, y0AbAB1 and so @F=@B1 � Aðsmax

A �y0

AkIÞ=y0

A40 (thelast inequality is due to the trivial requirementsmax

A 4y0AkI—otherwise, there are no steady states in whichA40). Thus, for small A values, @F=@B1 is positive and small.Therefore, close enough to the A¼0 plane, 9II9� 0, and thecondition a340 holds for SLH and SHL, but not for SLL and SHH.That is, close to the bifurcation points, the latter two steady statesare not stable. As we go further away from the A¼0 plane,however, @F=@B1 may become negative: For example, if weincrease A by means of increasing smax

A , A increases approximatelylinearly with this parameter (as can be seen from Eq. (5)), andtherefore the first term in rhs of Eq. (C.1) stays more or less fixed(as B1 remains approximately constant given such a change—see,for example, Fig. 5, left panels); and since the second term in therhs of Eq. (C.1) decreases linearly, @F=@B1 will indeed eventuallybecome negative. In order to understand the implications of this,we must inspect each steady state separately.

For SLH, @G=@Ao0. Thus, as long as @F=@B140 (and in parti-cular close to the bifurcation point), a340 for this steady state.However, when Ab0, @F=@B1 may become negative, as explainedabove. If its absolute value gets large enough, the condition a340will be violated, and the steady state will lose its local stability.This may give rise to a limit cycle around SLH, as is indeedexemplified in Figs. 5d and 7.

For SHL, @G=@A40. Therefore, for extremely small and extre-mely large A values (i.e., when @F=@B1 is positive but small, orwhen it is negative, respectively), the condition a340 holds forthis steady state. However, for certain intermediate A values, thissteady state may become unstable. Here, too, this may result in alimit cycle, as demonstrated in Fig. 7.

As explained above, for the other two steady states, a3o0close to the bifurcation points in which they merge with theirA ¼ 0 counterparts. In principle, for larger A values, it may bepossible that for these steady states the condition a340 holds.However, we did not encounter such cases in any numericalsimulation we ran.

Appendix D. Parameter values used in each figure

�
Fig. 2. (a) smaxA ¼ 2� 106, yA ¼ 104, dA ¼ 100, kI ¼ 5� 10�4, dI ¼
120. (b) sB2 ¼ 20, dB2 ¼ 0:5, pB2 ¼ 1:5, yB12 ¼ 103, yB22 ¼ 108.(c)–(d) sB1 ¼ 20, dB1 ¼ 0:5, pB1 ¼ 1:5, yB11 ¼ 103, yB21 ¼ 108,aA ¼ 10.
� Fig. 3. sB1 ¼ sB2 ¼ 10, dB1 ¼ dB2 ¼ 0:5, pB1 ¼ pB2 ¼ 1:5,yB11 ¼ yB12 ¼ 103, yB21 ¼ yB22 ¼ 106. � Fig. 4. In all panels, unless otherwise stated: sB1 ¼ sB2 ¼ 20,dB1 ¼ dB2 ¼ 0:5, pB1 ¼ pB2 ¼ 1:5, yB11 ¼ yB12 ¼ 103, yB21 ¼ yB22 ¼
108, aA ¼ 1, kI ¼ 5� 10�4. (a) smaxA ¼ 2� 10�1, yA ¼ 10�3,

dA ¼ 10�3, dI ¼ 120. (b) smaxA ¼ 8� 107, yA ¼ 104, dA ¼ 5� 10�4,

kI ¼ 5� 10�5, dI ¼ 120, sB1 ¼ sB2 ¼ 2. (c) smaxA ¼ 50, yA ¼ 1,

dA ¼ 1, dI ¼ 1. (d) smaxA ¼ 2� 106, yA ¼ 104, dA ¼ 5� 10�4,

dI ¼ 194, sB1 ¼ sB2 ¼ 2. (e) smaxA ¼ 2� 108, yA ¼ 106, dA ¼ 100,

dI ¼ 195. (f) smaxA ¼ 5� 108, yA ¼ 106, dA ¼ 100, dI ¼ 120.

(g) smaxA ¼ 4� 109, yA ¼ 107, dA ¼ 100, dI ¼ 199. (h) smax

A ¼

5� 1011, yA ¼ 106, dA ¼ 10, kI ¼ 5� 10�7, dI ¼ 100.
� Fig. 5. In all panels, sB1 ¼ sB2 ¼ 20, dB1 ¼ dB2 ¼ 0:5, pB1 ¼ pB2 ¼
1:5, yB11 ¼ yB12 ¼ 103, yB21 ¼ yB22 ¼ 108, aA ¼ 1. (a)–(b)dA ¼ 10�3, yA ¼ 10�3, kI ¼ 5� 10�4, dI ¼ 120. In (a) smax

A ¼ 0:2;in (b) smax

A ranges from 0.12 to 240.12. (c)–(d) dA ¼ 5� 10�4,yA ¼ 104, kI ¼ 5� 10�4, dI ¼ 100. In (c) smax

A ¼ 1:01� 106; in(d) smax

A ranges from 106 to 1:001� 109. (e)–(f) dA ¼ 100,yA ¼ 107, kI ¼ 5� 10�4, dI ¼ 199. In (e) smax

A ¼ 2� 109; in(f) smax

A ranges from 1:99� 109 to 3:98� 1017.
� Fig. 6. Unless otherwise stated, in both panels, sB1 ¼ 20,dB1 ¼ 0:5, pB1 ¼ 1:5, aA ¼ 1, yB11 ¼ 103, yB21 ¼ 108, smax
A ¼

2� 108, yA ¼ 106, dA ¼ 100, kI ¼ 1� 10�2, dI ¼ 80. (a) dI ¼ 180.(b) yB11 ¼ 104, yB21 ¼ 109.
� Fig. 7. In both panels, sB1 ¼ sB2 ¼ 20, dB1 ¼ dB2 ¼ 0:5, pB1 ¼
pB2 ¼ 1:5, aA ¼ 1. (a) smaxA ¼ 1:75� 107, yA ¼ 104, dA ¼ 5� 10�4,

kI ¼ 5� 10�4, dI ¼ 120, yB11 ¼ yB12 ¼ 103, yB21 ¼ yB22 ¼ 108.(b) smax

A ¼ 2� 108, yA ¼ 106, dA ¼ 1, kI ¼ 10�4, dI ¼ 198:5,yB11 ¼ yB12 ¼ 104, yB21 ¼ yB22 ¼ 106.
� Fig. 8. smax
A ¼ 2� 105, yA ¼ 103, dA ¼ 0:1, kI ¼ 5� 10�3, dI ¼ 50,sB1 ¼ sB2 ¼ 5, dB1 ¼ dB2 ¼ 0:5, pB1 ¼ pB2 ¼ 1:5, aA ¼ 1, yB12 ¼ 102,yB22 ¼ 104. In the case corresponding to a functional TLR9(panel (a), blue bars in panels (c)–(f)), yB11 ¼ 103, yB21 ¼ 105.Following the knockout of TLR9 (panel (b), red bars in panels(c)–(f)), yB11 ¼ 5� 104, yB21 ¼ 5� 105. In both cases, the initialvalues of the variables are Að0Þ ¼ 5,000, Ið0Þ ¼ 0, B1ð0Þ ¼ VB1 ¼

10, B2ð0Þ ¼ VB2 ¼ 10.
� Fig. 9. In both panels, smax
A ¼ 6� 108, yA ¼ 105, dA ¼ 0:1, kI ¼

5� 10�3, dI ¼ 50, sB1 ¼ sB2 ¼ 5, dB1 ¼ dB2 ¼ 0:5, pB1 ¼ pB2 ¼ 1:5,aA ¼ 1, yB12 ¼ 103, yB22 ¼ 108. (a) yB11 ¼ 103, yB21 ¼ 108.(b) yB11 ¼ 104, yB21 ¼ 5� 109.
� Fig. 10. smax
A ¼ 2� 105, yA ¼ 3,900, dA ¼ 0:1, kI1 ¼ 5� 10�4,kI2 ¼ 5� 10�5, dI ¼ 50, sB1 ¼ sB2 ¼ 5, dB1 ¼ dB2 ¼ 0:5, pB1 ¼

pB2 ¼ 1:5, aA1 ¼ 0:1, aA2 ¼ 0:05. In the case corresponding to afunctional TLR9 (panel (a), blue bars in panels (c)–(f)),yB11 ¼ 103, yB21 ¼ 105, yB12 ¼ 102, yB22 ¼ 104. Following theknockout of TLR9 (panel (b), red bars in panels(c)–(f)),yB11 ¼ 5� 104, yB21 ¼ 1� 106, yB12 ¼ 5� 102, yB22 ¼ 5� 104.In both cases, the initial values of the variables areAð0Þ ¼ 50,000, Ið0Þ ¼ 0, B1ð0Þ ¼ VB1 ¼ 10, B2ð0Þ ¼ VB2 ¼ 10.

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